Relation between solutions of the Schrödinger equation with transitioning resonance solutions of the gravitational three-body problem

The purpose of this paper is to describe a mechanism to globally model the solutions of a modified Schrödinger equation and how they transition between energy states, with a special set of approximate resonance solutions to the classical gravitational Newtonian three-body problem, for a wide range of masses. These resonance solutions transition from one resonance to another through the process of weak capture.

We consider a special version of the three-body problem that has proved to be useful in understanding the complexities of three-body motion, in the macroscopic scale, going back to Poincaré [1]. This is the circular restricted three body-problem, where the motion of one body, P₀, is studied as it moves under the influence of the gravitational field of ${P}_{1},{P}_{2}$, assumed to move in mutual circular orbits of constant frequency ω. It is also assumed that the mass of P₀, labeled m₀, is negligible with respect to the masses of ${P}_{1},{P}_{2}$, labeled ${m}_{1},{m}_{2}$, respectively. In this paper, we will also assume that m₂ is much smaller than m₁, ${m}_{2}\ll {m}_{1}$. For example, in the case of planetary objects, one can take ${P}_{1},{P}_{2}$ to be the Earth, Moon, respectively, and P₀ to be a rock. One can scale down ${m}_{0},{m}_{1},{m}_{2}$, as well as the relative distances between the particles, until the quantum scale is reached where the pure gravitational modeling is no longer sufficient. When using a rotating coordinate system that rotates about the center of mass of ${P}_{1},{P}_{2}$ with constant frequency ω, it is well known that Hamiltonian function for the motion of P₀ is time independent defining a conservative system (see section 3 ).

For a general class of conservative systems, that includes, for example, the restricted three-body problem considered here, it is known that such systems can be associated to the Schrödinger equation. As is described in Lanczos [2], this can be done by computing the action function $S(x)$ for the motion of P₀, where x is the position of P₀.¹ S is a solution of the Hamilton-Jacobi partial differential equation associated to the restricted three-body problem. The $x(t)$ are orthogonal to the surface $S(x)=C$, for constant C. In this sense, the iso-S surfaces locally determine the trajectories $x(t)$, that is for t having sufficiently small variation. On the other hand, it can be shown that S is the phase of wave solutions to the Schrödinger equation. Thus, $S(x)=C$ is a wave surface. Conversely, starting with the S satisfying the Schrödinger equation, one obtains the Hamiltonian-Jacobi equation provided the Planck's constant $\hslash \to 0$. This equivalence is local since in general S can only be shown to locally exist as a solution to the Hamiltonian-Jacobi equation, which is the case for the three-body problem. For the full equivalence it is necessary that one restricts $\hslash \to 0$ that is not realistic. More significantly, this equivalence does not determine the global behavior of the solutions $x(t)$ and how they dynamically can model the transition of energy states for solutions of the Schrödinger equation.

This paper will use methods of dynamical systems to globally determine special resonance motions for P₀ that are shown to be equivalent to different energy states for a modified Schrödinger equation and where the transition between energy states is equivalent to the process of weak capture in the three-body problem described in this paper. This result does not require the restriction of Planck's constant, and doesn't use Planck's constant in the modeling. The action function S is not used. Due to the complexity of the motions described, it is seen that using the iso-S surfaces to locally determine the global solutions would not seem feasible.

We describe a mechanism for the existence of a special family, ${\mathfrak{F}}$, of approximate resonance motions of P₀ about P₁, that transition from one resonance to another by the process of weak capture by P₂. This is a temporary capture defined in section 2. These motions are approximately elliptical with frequency ${\omega }_{1}\equiv {\omega }_{1}(m/n)$. ${\omega }_{1}(m/n)(t)$ or equivalently ${\omega }_{1}(t)$, are functions of the time t, where ${\omega }_{1}(m/n)$ is approximately equal to the constant values $(m/n)\omega$, $m,n$ are positive integers. That is, the period of motion of P₀ is approximately synchronized with the circular motion of P₂ about P₁, where in the time P₀ makes approximately n revolutions about P₁, P₂ makes m revolutions about P₁. The approximate resonance value of the frequency means that $| {\omega }_{1}(t)-(m/n)\omega | \lt \delta$, for a small tolerance δ as time varies for restricted time spans, described in section 2. When P₀ is moving on an approximate resonance orbit about P₁, it will eventually move away from this orbit and become captured temporarily about P₂, in weak capture. When P₀ escapes from this capture, it again moves about P₁ in another resonance elliptical orbit, with approximate resonance $m^{\prime} /n^{\prime}$. This process repeats either indefinitely, or ends when, for example, P₀ escapes the ${P}_{1},{P}_{2}$-system. This also implies that the approximate two-body energy E₁ of P₀ about P₁ can only take on a discrete set of values, ${E}_{1}(m/n)(t)$ at each time t, which are approximately constant defined by the resonances. This is stated as result A in section 2 and as theorem A in section 3. The properties and dynamics of weak capture, and weak escape, are described section 3. Comets can perform such resonance transitions(see sections 2 3).

A modified Schrödinger equation is defined for the motion of P₀ about P₁, under the gravitational perturbation of P₂. This is first considered in the case of macroscopic masses. It is given by,

$$\begin{eqnarray}&&-\displaystyle \frac{{\sigma }^{2}}{2\nu }{{\rm{\nabla }}}^{2}{\rm{\Psi }}+\bar{V}{\rm{\Psi }}=E{\rm{\Psi }},\end{eqnarray} \tag{ 1 }$$

where ${{\rm{\nabla }}}^{2}\equiv {\rm{\nabla }}\cdot {\rm{\nabla }}$ is the Laplacian operator, $\bar{V}$ is an averaged three-body gravitational potential, E is the energy, and ν is the reduced mass for ${P}_{0},{P}_{1}$. σ is a function that depends on ${m}_{0},{m}_{1}$ and G, the gravitational constant. σ replaces $\hslash =h/2\pi$, h is Planck's constant that is in the classical Schrödinger equation. In this case, for macroscopic values of the masses, since P₀ is not a wave, Ψ is used to determine the probability distribution function, $| {\rm{\Psi }}{| }^{2}$, of locating P₀ near P₁ as a macroscopic body. We show in section 4 that E can only take on the following approximate quantized values,

$$\begin{eqnarray}&&{E}_{\tilde{n}}=-\displaystyle \frac{4\sigma }{{\tilde{n}}^{2}},\end{eqnarray} \tag{ 2 }$$

$\tilde{n}=1,2,3,\ldots$. As seen in section 2, this implies that the frequency of P₀ takes a particularly simple form that is independent of any parameters. These frequencies have the approximate values, $8/{\tilde{n}}^{3}$. Ψ is explicitly computed in section 4. $| {\rm{\Psi }}{| }^{2}$ is shown to be exponentially decreasing as a function of the distance of P₀ from P₁. The general solution, Ψ, of the modified Schrödinger equation is described in result B in section 2.

A main result of this paper is that the quantized energy values ${E}_{\tilde{n}}$ correspond to a subset, ${\mathfrak{U}}$, of the resonance orbit family, ${\mathfrak{F}}$, of P₀ about P₁. This is listed as result C in section 2. This provides a global equivalence of the solutions of the modified Schrödinger equation with the transitioning resonance solutions of the the three-body problem.

As a final result, we show is that the solution, Ψ, for the location of P₀ for the modified Schrödinger equation for the macroscopic values of the masses, can be extended into the quantum-scale. This is summarized as result D in section 2. This gives a way to mathematically view the resonance motions in the quantum-scale, as an extension of the resonance solutions for macroscopic particles. Other models, such as the classical Schrödinger and Schrödinger-Newton equations are given in latter sections.

The results of this paper are described in detail and summarized in section 2. This section contains the main findings of this paper. Additional details, derivations, and proofs are contained in section 3 for weak capture and in section 4 for the modified Schrödinger equation.

In this section we elaborate on the results described in the Introduction. The first set of results pertain to a family of resonance orbits about P₁ obtained from the three-body problem and the second set of results pertain to finding these orbits using a modified Schrödinger equation.

2.1. Resonance orbits in the three-body problem and weak capture

The motion of P₀ is defined for the circular restricted three-body problem described in the Introduction. It is sufficient to use the planar version of this model, without loss of generality for the purposes of this paper, where P₀ moves in the same plane of motion as that of the uniform circular motion of ${P}_{1},{P}_{2}$ of constant frequency ω (see section 3). The macroscopic masses satisfy, ${m}_{2}/{m}_{1}\ll 1$ and the mass of m₀ is negligibly small so that P₀ does not gravitationally perturb ${P}_{1},{P}_{2}$, but ${P}_{1},{P}_{2}$ perturb the motion of P₀. We consider an inertial coordinate system, $({X}_{1},{X}_{2})\in {{\mathbb{R}}}^{2}$, whose origin is the center of mass of ${P}_{1},{P}_{2}$.

The differential equations for P₀ are given by the classical system

$$\begin{eqnarray}&&\ddot{X}={{\rm{\Omega }}}_{X}(X,t),\end{eqnarray} \tag{ 3 }$$

where $X=({X}_{1},{X}_{2})\in {{\mathbb{R}}}^{2}$, $t\in {{\mathbb{R}}}^{1}$, ${}^{\cdot }\equiv \displaystyle \frac{d}{{dt}}$, ${{\rm{\Omega }}}_{X}\equiv ({{\rm{\Omega }}}_{{X}_{1}},{{\rm{\Omega }}}_{{X}_{2}})$ (${{\rm{\Omega }}}_{X}\equiv \partial {\rm{\Omega }}/\partial X$) and

$$\begin{eqnarray}&&{\rm{\Omega }}=\displaystyle \frac{{{Gm}}_{1}}{{r}_{1}(t)}+\displaystyle \frac{{{Gm}}_{2}}{{r}_{2}(t)}\end{eqnarray} \tag{ 4 }$$

where ${r}_{1}(t)=| X-{a}_{1}(t)|$, ${r}_{2}=| X-{a}_{2}(t)|$, $| \cdot |$ is the standard Euclidean norm. The mutual circular orbits of ${P}_{1},{P}_{2}$ are given by ${a}_{1}(t)={\rho }_{1}(\cos {\omega }_{a}t,\sin {\omega }_{a}t)$, ${a}_{2}(t)=-{\rho }_{2}(\cos {\omega }_{b}t,\sin {\omega }_{b}t)$, with constant circular frequencies ${\omega }_{a},{\omega }_{b}$ of ${P}_{1},{P}_{2}$, respectively. We have divided both sides of (3) by m₀ and then took the limit as ${m}_{0}\to 0$. It is well known that the solutions of the circular restricted three-body problem for P₀ accurately model the motion of P₀ in the general three-body problem for circular initial conditions for ${P}_{1},{P}_{2}$, with m₀ kept positive and negligibly small.

It is noted that all solutions $\xi (t)=(X(t),\dot{X}(t))\in {{\mathbb{R}}}^{4}$ considered in this study will be ${C}^{\infty }$ in both t and initial conditions, $\xi ({t}_{0})=(X({t}_{0}),\dot{X}({t}_{0}))$ at an initial time t₀. We refer to ${C}^{\infty }$ as smooth dependence. More exactly, this means that all derivatives of $\xi (t)$ with respect to t of all orders are continuous and all partial derivatives of $\xi (t,\xi ({t}_{0}))$ with respect to ${X}_{1}({t}_{0}),{X}_{2}({t}_{0}),{\dot{X}}_{1}({t}_{0}),{\dot{X}}_{2}({t}_{0})$, of all orders, are continuous.

Although m₀ is taken in the limit to be 0 in the definition of the differential equations for the motion of P₀, we will assume it is non-zero but still negligible in mass with respect ${P}_{1},{P}_{2}$, ${m}_{0}\gtrapprox 0$, in all equations that follow.

We transform to a P₁-centered coordinate system for the restricted three-body problem. In this system, P₂ moves about P₁ at a constant distance β, with constant circular frequency $\omega =\sqrt{G({m}_{1}+{m}_{2})/\beta }$. Before stating our first result, two definitions are needed.

Assuming m₂ is much smaller than m₁, when P₀ moves about P₁ with elliptic initial conditions at an initial time $t={t}_{0}$, this elliptic motion will be slightly perturbed by P₂ ² . Let a₁ be the semi-major axis of P₀ with respect to P₁. As a function of time, a₁ will vary. If ${m}_{2}=0$, then a₁ is constant since P₀ will move on a pure ellipse. If m₂ is small, then P₀ moves in a nearly elliptic orbit about P₁, and ${a}_{1}(t)$ will be nearly constant for restricted time spans. This orbital element, along with the eccentricity, ${e}_{1}(t)$, true anamoly, ${\theta }_{1}(t)$, and other orbital elements, can be calculated for each t using the variational differential equations obtained from (3).(see [3–6]). These are referred to as osculating elements. e₁ will likewise be nearly constant for a nearly elliptical orbit of P₀ anout P₁.

The variation of the frequency ${\omega }_{1}(t)$ can be obtained from ${a}_{1}(t)$: The osculating two-body period, T₁, of P₀ is explicitly related to ${a}_{1}(t)$ by Kepler's Third Law, ${a}_{1}^{3}={\left(2\pi \right)}^{-2}{T}_{1}^{2}G({m}_{0}+{m}_{1})$, and $\omega (t)={T}_{1}^{-1}$.

Definition 1. An approximate resonance orbit, ${{\rm{\Phi }}}_{m/n}(t)$, of P₀ moving about P₁ in a P₁-centered coordinate system, $Y=({Y}_{1},{Y}_{2})$, as a function of t in resonance with P₂, is an approximate elliptical orbit of frequency ${\omega }_{1}={\omega }_{1}(t)$, where ${\omega }_{1}\approx (m/n)\omega$. $m,n$ are positive integers. Thus, ${\omega }_{1}$ is approximately constant as time varies. In phase space, $(Y,\dot{Y})\in {{\mathbb{R}}}^{4}$, ${{\rm{\Phi }}}_{m/n}(t)=({Y}_{1}(t),{Y}_{2}(t),{\dot{Y}}_{1}(t),{\dot{Y}}_{2}(t))$. ${{\rm{\Phi }}}_{m/n}(t)$ has a period ${T}_{1}={\omega }_{1}^{-1}$, approximately constant. ${T}_{1}\approx (n/m)T$, T is the constant circular period of P₂ about P₁, $T={\omega }^{-1}$. For notational purposes, we refer to an approximate resonance orbit as a resonance orbit for short. A resonance orbit with ${\omega }_{1}\approx (m/n){\boldsymbol{\omega }}$ is also referred to as a $n:m$ resonance orbit. (Nearly resonant motion, related to approximate resonance motion, is described in [7].)

The term 'approximate' in definition 1 means to within a small tolerance, ${ \mathcal O }(\delta )$, $\delta ={m}_{2}/{m}_{1}\ll 1$. ${ \mathcal O }(\delta )$ is a function of time, t, and smooth in t. An approximate elliptic orbit means that the variation of the orbital parameters(${\omega }_{1}$, a₁, e₁) of ${\rm{\Phi }}(t)=(Y(t),\dot{Y}(t))$, with respect to P₁ in a P₁-centered coordinate system, will slightly vary by ${ \mathcal O }(\delta )$ due to the gravitational perturbation of P₂. The two-body energy of ${{\rm{\Phi }}}_{m/n}(t)$ with respect to P₁ is labeled, ${E}_{1}(m/n)$ =${E}_{1}(m/n)(t)$, which is approximately constant. Note that the general Kepler energy ${E}_{1}(t)$ along a trajectory $(Y(t),\dot{Y}(t)$ is given by (7).

Thus, ${\omega }_{1}\approx (m/n)\omega$ is equivalent to ${\omega }_{1}=(m/n)\omega +{ \mathcal O }(\delta )$. ${a}_{1},{e}_{1}$ likewise vary within a variation ${ \mathcal O }(\delta )$. The variations ${ \mathcal O }(\delta )$ are all different functions for the different parameters, but the same notation is used. The tolerance on these orbital parameters is valid for finite times. We assume that t varies over finite time spans. Thus, for a given variable, say ${\omega }_{1}(t)$, if ${\epsilon }_{1}\gt 0$ is a given number, and $t\in [{t}_{0},{t}_{1}]$, ${t}_{1}\gt {t}_{0}$, m₂ can be taken small enough so that $| { \mathcal O }(\delta )| \lt {\epsilon }_{1}$

For a resonance orbit to be well defined, it is assumed that ${m}_{2}\gt 0$. If ${m}_{2}=0$, then even though ω is defined, P₂ no longer exists. Thus, we assume ${m}_{2}\gt 0$ throughout this paper, unless otherwise indicated. This assumption is also necessary for the definition of weak capture.

We define 'weak capture' of P₀ about P₂. In this case, we change to a P₂-centered coordinate system. This type of capture is discussed in section 3. Weak capture is where the two-body energy, E₂, of P₀ with respect to P₂ is temporarily negative. It is used to define an interesting region about P₂ described in the next section called the weak stability boundary. Chaotic motion occurs on and near this region.

Definition 2. P₀ has weak capture about P₂, in a P₂-centered coordinate system, $Z=({Z}_{1},{Z}_{2})$, at a time t₀ if the two-body energy, E₂, of P₀ with respect to P₂, is negative at t₀ and for a finite time after where it becomes positive(P₀ escapes). More precisely, E₂ is given by

$$\begin{eqnarray}&&{E}_{2}=\displaystyle \frac{1}{2}| \dot{Z}{| }^{2}-\displaystyle \frac{G({m}_{0}+{m}_{2})}{{r}_{2}},\end{eqnarray} \tag{ 5 }$$

${r}_{2}=| Z| \gt 0$. Let ${ \mathcal Z }(t)=({Z}_{1}(t),{Z}_{2}(t),{\dot{Z}}_{1}(t),{\dot{Z}}_{2}(t))$ be a solution for the differential equations (3) in P₂-centered coordinates for $t\geqslant {t}_{0}$. P₀ is weakly captured at t₀ if ${E}_{2}({ \mathcal Z }(t))\lt 0$ for ${t}_{0}\leqslant t\lt {t}_{1}$, ${t}_{0}\lt {t}_{1}$, ${E}_{2}({ \mathcal Z }({t}_{1}))=0$, ${E}_{2}({ \mathcal Z }(t))\gtrsim 0$ for $t\gtrsim {t}_{1}$. After P₀ leaves weak capture at t₁, we say that P₀ has weak escape from P₂ at $t={t}_{1}$. Weak capture in backwards time from t₀ is similarly defined.

P₀ is captured at a point $({Z}_{1}(t),{Z}_{2}(t))$ of a trajectory ${ \mathcal Z }(t)$ at ${t}^{* }$ if ${E}_{2}({ \mathcal Z }({t}^{* }))\lt 0$. Capture at a point need not imply weak capture, in forward time, since P₀ could be captured for all time $t\gt {t}^{* }$.

This dynamics is summarized in result A and proven in section 3, where it is formulated more precisely as Theroem A.

Result A. Weak capture of P₀ about P₂ at a time t₀ yields resonance motion of P₀ about P₁, which repeats yielding a family, ${\mathfrak{F}}$, of resonance orbits. More precisely,

Assume P₀ is weakly captured by P₂ at time $t={t}_{0}$, then

• (i)  
  As t increases from t₀, P₀ first escapes from P₂ and then P₀ moves onto a resonance orbit, ${{\rm{\Phi }}}_{m/n}(t)$, about P₁. P₀ performs a finite number of cycles about P₁ until it eventually moves again to weak capture by P₂, where the process continues and P₀ moves onto another resonance orbit. When P₀ moves from ${{\rm{\Phi }}}_{m/n}(t)$ to another resonance orbit, ${{\rm{\Phi }}}_{m^{\prime} /n^{\prime} }(t)$, $m^{\prime} ,n^{\prime}$ may or may not equal $m,n$. In general, a sequence of resonance orbits is obtained, $\{{{\rm{\Phi }}}_{m/n}(t),{{\rm{\Phi }}}_{m^{\prime} /n^{\prime} }(t),\ldots \}$. The process stops when P₀ escapes the ${P}_{1}{P}_{2}$-system, collides with P₂ or moves away from a resonance frequency. This set of resonance orbits forms a family, ${\mathfrak{F}}$, of orbits, that depend on the initial weak capture condition.
• (ii)  
  When P₀ moves onto a sequence of resonance orbits about P₁ as described in (i), then a discrete set of energies are obtained, $\{{E}_{1}(m,n),{E}_{1}(m^{\prime} ,n^{\prime} ),\ldots \}$.

The transitioning of ${{\rm{\Phi }}}_{m/n}(t)$ to ${{\rm{\Phi }}}_{m^{\prime} /n^{\prime} }(t)$ is shown in a sketch in figure 1.

Zoom In Zoom Out Reset image size

The proof of theorem A is given in detail in section 3.

Applications of theorem A a to comet motions and numerical simulations is described in section 3.

A key result obtained in section 3 is,

Lemma A. The frequency ${\omega }_{1}(m/n)(t)$ of ${{\rm{\Phi }}}_{m/n}(t)$ is given by,

$$\begin{eqnarray}&&{\omega }_{1}(m/n)=(m/n)\omega +{ \mathcal O }(\delta ),\end{eqnarray} \tag{ 6 }$$

where ${ \mathcal O }(\delta )(t)$ is smooth in t.

(i) Is proven in section 3 as theorem A. We prove (ii): consider the general two-body energy of P₀ with respect to P₁. It is given by,

$$\begin{eqnarray}&&{E}_{1}\equiv {E}_{1}(Y,\dot{Y})=\displaystyle \frac{1}{2}| \dot{Y}{| }^{2}-\displaystyle \frac{G({m}_{0}+{m}_{1})}{{r}_{1}},\end{eqnarray} \tag{ 7 }$$

where $Y=({Y}_{1},{Y}_{2})$, ${r}_{1}=| Y|$, are inertial P₁-centered coordinates. E₁ can be written as ${E}_{1}=-G({m}_{0}+{m}_{1})/(2{a}_{1})$, [8]. Using Kepler's third law relating the period, T₁, to a₁, implies, ${E}_{1}=-A{\omega }_{1}^{2/3}$, $A=(1/2){\left(2\pi G({m}_{0}+{m}_{1}\right)}^{2/3}$. (6) Implies E₁ can be written as,

$$\begin{eqnarray}&&{E}_{1}({{\rm{\Phi }}}_{m/n}(t))\equiv {E}_{1}(m/n)=-{[(m/n)\omega ]}^{2/3}A+{ \mathcal O }(\delta ),\end{eqnarray} \tag{ 8 }$$

where the remainder is smooth in t. (8) Proves (ii). It is noted that in this case, ${E}_{1}\approx -{[(m/n)\omega ]}^{2/3}A$.

It is noted that E₁ can be written in an equivalent form to (7) by multiplying both sides of (7) by the reduced mass, $\nu ={m}_{0}{m}_{1}{\left({m}_{0}+{m}_{1}\right)}^{-1}$, yielding

$$\begin{eqnarray}&&{\tilde{E}}_{1}\equiv \nu {E}_{1}=\displaystyle \frac{\nu }{2}| \dot{Y}{| }^{2}-\displaystyle \frac{{{Gm}}_{0}{m}_{1}}{{r}_{1}},\end{eqnarray} \tag{ 9 }$$

which implies, ${\tilde{E}}_{1}=-{{Gm}}_{0}{m}_{1}/(2{a}_{1}).$ This scaled energy is more convenient to use when the modified Schrödinger equation is considered. To obtain the corresponding two-body differential equations with (9) as an integral, one multiplies the differential equations associated to (7) by ν. The solutions are the same for both sets of differential equations³ . Thus, it is seen that ${\tilde{E}}_{1}=\nu {E}_{1}=-\nu A{\omega }_{1}^{2/3}$. Setting $\sigma =\nu A$ implies,

$$\begin{eqnarray}&&{\tilde{E}}_{1}=-\sigma {\omega }_{1}^{2/3},\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&\sigma =(1/2){\left(2\pi G\right)}^{2/3}{m}_{0}{m}_{1}{\left({m}_{0}+{m}_{1}\right)}^{-1/3}.\end{eqnarray} \tag{ 11 }$$

${\omega }_{1}$ remains the same since the solutions haven't changed. This defines the function σ that plays a key role in this paper.

2.2. A modified Schrödinger equation: macroscopic scale

Analogy A. It is noted that equation (10) has a form similar to the Planck-Einstein relation for quantum mechanics for the energy, ${ \mathcal E }$, of a photon, ${ \mathcal E }=h\lambda$, where λ is the frequency of the photon. σ is analogous to h and ${\tilde{{\omega }_{1}}}^{2/3}$ is analogous to λ. There is another analogy for the case of an electron, P₀, moving about an atomic nucleus: when P₀ changes from one orbital to another, the energy of the photon absorbed or emitted is given by ${\rm{\Delta }}E=h\lambda$, where ${\rm{\Delta }}E={E}_{1}-{E}_{2}$, where E_i is the energy of P₀ in the ith orbital, $i=1,2$. This process is analogous to a macroscopic particle P₀ changing from from one resonance orbit ${{\rm{\Phi }}}_{m/n}(t)$ in ${\mathfrak{F}}$ to another, ${{\rm{\Phi }}}_{m^{\prime} /n^{\prime} }(t)$, through weak capture and escape, where ${\rm{\Delta }}{\tilde{E}}_{1}={\tilde{E}}_{1}(m/n)-{\tilde{E}}_{1}(m^{\prime} /n^{\prime} )$. ${\rm{\Delta }}{\tilde{E}}_{1}$ is analogous to ${\rm{\Delta }}E$.

We consider a modified Schrödinger equation (1). This Schrödinger equation differs from the classical one by replacing $\hslash$ by the function $\sigma =\sigma ({m}_{0},{m}_{1},G)$ and V by a three-body potential $\hat{V}$ derived from the circular restricted three-body problem. The choice of σ is motivated by analogy A. In this modeling, the masses ${m}_{0},{m}_{1},{m}_{2}$ and distance between them is assumed to not be in the quantum scale. Under this assumption, Ψ does not represent a wave motion and is used to measure the probability of locating P₀ at a distance r₁ from P₁.

The potential $\bar{V}$ in (1), is derived from the restricted three-body problem modeling for the motion of P₀. In inertial coordinates, $Y=({Y}_{1},{Y}_{2})$ centered at P₁, P₂ moves about P₁ on a circular orbit, $\gamma (t)$ of constant radius β and circular frequency $\omega =\sqrt{G({m}_{1}+{m}_{2})/\beta }$, $\gamma (t)=\beta (\cos \omega t,\sin \omega t)$. The potential for P₀ is given by

$$\begin{eqnarray}&&\hat{V}={V}_{1}+{V}_{2},\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{V}_{1}=-\displaystyle \frac{{{Gm}}_{0}{m}_{1}}{r},\hspace{.2in}{V}_{2}=-\displaystyle \frac{{{Gm}}_{0}{m}_{2}}{{r}_{2}},\end{eqnarray} \tag{ 13 }$$

where $r=| Y| ,{r}_{2}=| Y-\gamma (t)|$. For simplicity of notation, we have replaced the symbol r₁ by r (${r}_{1}\equiv r)$.

We replace V₂ by an approximation given by an averaged value of V₂ over one cycle of $\gamma (t)$, $t\in [0,2\pi /\omega ]$,

$$\begin{eqnarray}&&{\bar{V}}_{2}=-\displaystyle \frac{{{Gm}}_{0}{m}_{2}\omega }{2\pi }{\int }_{0}^{\tfrac{2\pi }{\omega }}\displaystyle \frac{{dt}}{\sqrt{{r}^{2}+{\beta }^{2}-2\beta ({Y}_{1}\cos \omega t+{Y}_{2}\sin \omega t)}}.\end{eqnarray} \tag{ 14 }$$

It is proven in section 4 that ${\bar{V}}_{2}$ can be reduced to three cases in (48) depending om $r\lt 0,r=0,r\gt 0$, respectively, where the first order term in ${\bar{V}}_{2}$ has a form similar to V₁.

As an approximation to $\hat{V}$ we use,

$$\begin{eqnarray}&&\bar{V}={V}_{1}+{\bar{V}}_{2}.\end{eqnarray} \tag{ 15 }$$

The modified Schrödinger equation that we consider is given by

$$\begin{eqnarray}&&-\displaystyle \frac{{\sigma }^{2}}{2\nu }{{\rm{\nabla }}}^{2}{\rm{\Psi }}+\bar{V}{\rm{\Psi }}=E{\rm{\Psi }},\end{eqnarray} \tag{ 16 }$$

The solution of this modified Schrödinger equation is derived in section 4. The solution is summarized in result B.

Result B. The explicit solution of the modified Schrödinger equation, more generally in a three-dimensional P₁-centered inertial coordinates, $({Y}_{1},{Y}_{2},{Y}_{3})$, (16), is given by,

$$\begin{eqnarray}&&{\rm{\Psi }}={R}_{\tilde{n},l}(r){Y}_{{m}_{l},l}(\phi ,\theta )+{ \mathcal O }({m}_{0}{m}_{2}).\end{eqnarray} \tag{ 17 }$$

$\tilde{n}=0,1,2\ldots$; $l=0,1,2,\ldots$; $-l\leqslant {m}_{l}\leqslant l$, r is the distance from P₀ to P₁, $r\geqslant 0$, $\phi \in [0,2\pi ]$ is the angle in the ${Y}_{1},{Y}_{2}$-plane relative to the Y₁-axis, $\theta \in [0,\pi ]$ is the angle relative to the Y₃-axis. ${R}_{\tilde{n},l}(r)$ is given by (39) defined using Laguerre polynomials. ${Y}_{l,{m}_{l}}={{\rm{\Phi }}}_{{m}_{l}}(\phi ){{\rm{\Theta }}}_{l,{m}_{l}}(\theta )$ are spherical harmonic functions, where ${{\rm{\Phi }}}_{{m}_{l}}(\phi ),{{\rm{\Theta }}}_{l,{m}_{l}}(\theta )$ are given by (35), (37), respectively. ${\rm{\Theta }}(\theta )$ is defined by Legendre polynomials.

$| {\rm{\Psi }}{| }^{2}$ is the probability distribution function of finding P₀ at a location $(r,\phi ,\theta )$. In particular, the radial probability distribution function of finding P₀ at a radial distance r is given by

$$\begin{eqnarray}&&P(r)={R}^{2}(r){r}^{2}+{ \mathcal O }({m}_{0}{m}_{2}),\end{eqnarray} \tag{ 18 }$$

where $R\equiv {R}_{\tilde{n},l}$.

Ψ exists provided the energy, E, is quantized as,

$$\begin{eqnarray}&&E\equiv {\hat{E}}_{\tilde{n}}=-\displaystyle \frac{4\sigma }{{\tilde{n}}^{2}}+{ \mathcal O }({m}_{0}{m}_{2}),\end{eqnarray} \tag{ 19 }$$

where the remainder term is smooth in ${Y}_{1},{Y}_{2},{Y}_{3}$.

It is noted that the solution of (16) is valid for $\bar{V}=0$. However, we assume $\bar{V}\ne 0$ to compare with the resonance solutions of three-body problem, where ${Y}_{3}=0$. All the terms ${ \mathcal O }({m}_{0}{m}_{2})$ are smooth in ${Y}_{1},{Y}_{2},{Y}_{3}$.

It is assumed that m₀ can be taken sufficiently small, such that for any given small number, ${\epsilon }_{2}\gt 0$, and for $({Y}_{1},{Y}_{2},{Y}_{3})\in D$, D compact, the term ${ \mathcal O }({m}_{0}{m}_{2})$ in (19) satisfies, $| { \mathcal O }({m}_{0}{m}_{2})| \lt {\epsilon }_{2}$. In this sense, ${\hat{E}}_{\tilde{n}}\approx -\tfrac{4\sigma }{{\tilde{n}}^{2}}$.

The planar case is now assumed, ${Y}_{3}=0$, unless otherwise indicated.

Assumptions 1. The use of the approximate symbol for ${\hat{E}}_{\tilde{n}}$ using ${ \mathcal O }({m}_{0}{m}_{2})$, is different to the one given in definition 1, for ${\omega }_{1}$ using ${ \mathcal O }(\delta )$, $\delta ={m}_{2}/{m}_{1}$. In definition 1, ${ \mathcal O }(\delta )$ depends on t, and to bound it by a given small number ${\epsilon }_{1}$, t varies on a compact set and m₂ is taken sufficiently small. In the second case, ${ \mathcal O }({m}_{0}{m}_{2})$, depends on $({Y}_{1},{Y}_{2})$, and to bound it by a small number ${\epsilon }_{2}$, $({Y}_{1},{Y}_{2})$ varies on a compact set D and m₀ is taken sufficiently small. To satisfy both cases, it is necessary to assume ${m}_{0},{m}_{1}$ are sufficiently small. The use of $\approx$ is taken from context.

2.3. Equivalence of quantized energies with resonance solutions

The quantized values of the energy, ${\hat{E}}_{\tilde{n}}$, (19), are for the modified Schrödinger equation, (16). These energies are not obtained for the three-body problem, but result from an entirely different modeling. When they are substituted for ${\tilde{E}}_{1}$ in the two-body energy relation, (10), for P₀ moving about P₁, it is calculated in section 4, (53), that they yield rational values for the two-body frequency, ${\omega }_{1}$,

$$\begin{eqnarray}&&{\omega }_{1}{| }_{{\tilde{E}}_{1}={\hat{E}}_{\tilde{n}}}\equiv {\omega }_{1}(\tilde{n})=\displaystyle \frac{8}{{\tilde{n}}^{3}}+{ \mathcal O }({m}_{0}{m}_{2}).\end{eqnarray} \tag{ 20 }$$

It is significant that the leading dominant term of ${\omega }_{1}(\tilde{n})$ is independent of masses and distances. This implies that to first order the frequencies do not depend on the masses or any other parameters.

Thus, for $\tilde{n}=1,2,\,\ldots \,$, infinitely many frequencies are obtained, ${\omega }_{1}(\tilde{n})$. We would like to show that these frequencies correspond to $\tilde{n}$ resonance orbits of ${\mathfrak{F}}$ for the three-body problem. Hence, we need to compare the frequencies ${\omega }_{1}(\tilde{n})$, given by (20), with the frequencies ${\omega }_{1}(m/n)$, defined by (6). The following result is obtained,

Result C. The quantized energy values, (19), of the modified Schrödinger equation can be put into a one to one correspondence with a subset, ${\mathfrak{U}}$, of the resonance solutions ${{\rm{\Phi }}}_{m/n}(t)\in {\mathfrak{F}}$ of the circular restricted three-body problem, where

$$\begin{eqnarray}&&{\mathfrak{U}}=\{{{\rm{\Phi }}}_{m/n}(t)| m=8,n={\tilde{n}}^{3},\tilde{n}=1,2,\ldots \}.\end{eqnarray} \tag{ 21 }$$

This is proven in section 4 by scaling the restricted three-body problem and using the fact that this scaling does not effect the leading order term $\tfrac{8}{{\tilde{n}}^{3}}$ of ${\omega }_{1}(\tilde{(}n))$.

It is noted that $m=8,n={\tilde{n}}^{3}$ implies that in the time it takes P₀ to make ${\tilde{n}}^{3}$ cycles about P₁, P₂ makes 8 cycles about P₁.

As previously noted, the limiting case of ${m}_{2}=0$ has been excluded in this paper since it is degenerate in the sense that the resonance families of solutions no longer exist. One can make a comparison with quantized two-body elliptic orbits of P₀ about P₁ with the classical Schrödinger equation for ${m}_{2}=0$ (see [9], page 263), but this case does not yield the transitioning resonance solutions described in this paper.

2.4. Quantum scale

In this section we show how to prove result A. The idea of the proof of result A is to utilize the geometry of the phase space about P₁, P₂, where the motion of P₀ is constrained by Hill's regions. Within the Hill's regions, the dynamics associated to weak capture from near P₂ together with the global properties of the invariant hyperbolic manifolds around P₁ will yield the proof.

The planar circular restricted three-body problem in inertial coordinates is defined in section 2 by (3) for the motion of P₀. If P₀ moves about P₁ with elliptic initial conditions, and a rotating coordinate system is assumed that rotates with the same constant circular frequency ω between P₁ and P₂, then the motion is understood by the Kolmogorov-Arnold-Moser(KAM) theorem [1, 8]. It says that nearly all initial elliptic initial conditions of P₀ with respect to P₁ give rise to quasi-periodic motion, of the two frequencies, ${\omega }_{1},\omega$, where ${\omega }_{1}$ is the frequency of the elliptic motion of P₀ about P₁, provided they satisfy the condition that ${\omega }_{1}/\omega$ is sufficiently non-rational. For the relatively small set of motions of P₀ where ${\omega }_{1}/\omega$ is sufficiently close to a rational number, the motion is chaotic. It is also necessary to assume that $\mu ={m}_{2}/({m}_{1}+{m}_{2})$ is sufficiently small.

The planar modeling is assumed without loss of generality. This follows since the resonance orbits we will be considering for P₀ moving about P₁ are approximately two-body in nature. This implies approximate planar motion. These same orbits result from weak capture conditions and escape, which imply that the plane of motion of P₀ about P₁ will approximately be the same plane of motion as that of P₂ about P₁. Thus, co-planar modeling assumed in the restricted three-body problem is a reasonable assumption.

Whereas the motion of P₀ about P₁ is well understood by the KAM theorem for small m₂, the general motion of P₀ about P₂ is not well understood since it's considerably more unstable. The instability arises due to the fact that m₂ is much smaller than m₁, and the KAM theorm cannot be easily applied unless P₀ moves infinitely close to P₂ [12]. This implies that if P₀ starts with an initial two-body elliptic state with respect to P₂, its trajectory is substantially perturbed by the gravitational effect of P₁. The resulting motion of P₀ about P₂ is unstable and generally rapidly deviates from the initial elliptic state. Numerical simulations show the motion to be chaotic in nature. Results described in this section provide a way to better understand motion about P₂.

The notion of weak capture (defined in section 2) of P₀ about P₂ is useful in trying to understand the motion of P₀ about P₂ with initial elliptic conditions. The idea is to numerically propagate trajectories of the three-body problem with initial conditions that have negative energy, ${E}_{2}\lt 0$, with respect to P₂, and measure how they cycle about P₂, described in more detail later in this section. Generally, if P₀ performs k complete cycles about P₂, relative to a reference line emanating from P₂, without cycling about P₁, then the motion of P₂ is called 'stable', provided it returns to the line with ${E}_{2}\lt 0$, while if does not return to the line after $(k-1)$ complete cycles, and cycles about P₁, the motion is called 'unstable'. It is also called unstable if P₀ does return to the line, but where ${E}_{2}\gt 0$. (see figure 4) The line represents a two-dimensional surface of section in the four-dimensional phase space, ${{\mathbb{R}}}^{4}$. The set of all stable points about P₂ defines the 'kth stable set', ${W}_{k}^{s}$, and the set of all unstable points is called the 'kth unstable set', ${W}_{k}^{u}$. Points that lie on the boundary between ${W}_{k}^{s}$ and ${W}_{k}^{u}$ define a set, ${{ \mathcal W }}_{k}$, called the 'kth weak stability boundary'. The boundary points are determined algorithmically, by iterating between stable and unstable points [13].

Points that belong to ${W}_{k}^{u}$ are in weak capture with respect to P₂ since they start with ${E}_{2}\lt 0$, which lead to escape with ${E}_{2}\gt 0$ (proposition 3.1). However, this may not be the case for points in ${W}_{k}^{s}$ since after they cycle about P₂ k times, it is possible they can remain moving about P₂ for all future time and E₂ will be negative each time P₀ intersects the line.

${{ \mathcal W }}_{k}$ was first defined in [14], for the case $k=1$. This set has proved to have important applications in astrodynamics to enable spacecraft to transfer to the Moon and automatically go into weak capture about the Moon, that requires no fuel for capture. This was a substantial improvement to the Hohmann transfer, which requires substantial fuel for capture [8, 15] ⁴ . It also has applications in astrophysics on the Lithopanspermia Hypothesis [16]. The weak stability boundary was generalized to k-cycles, $k\gt 1$, with new details about its geometric structure in [17]. [13] makes an equivalence of ${{ \mathcal W }}_{k}$ with the stable manifolds of the Lyapunov orbits associated to collinear Lagrange points.

${W}_{k}^{s}$, ${W}_{k}^{u}$, ${{ \mathcal W }}_{k}$ are defined more precisely: we transform from $X=({X}_{1},{X}_{2})$ defined in (3) to a rotating coordinate system, $x=({x}_{1},{x}_{2})$, that rotates with frequency ω, so that in this system, ${P}_{1},{P}_{2}$ are fixed on the x₁-axis. Scaling ${m}_{1}=1-\mu ,{m}_{2}=\mu ,\mu \gt 0,G=1,\beta =1,\omega =1$, as mentioned in section 2, we place P₁ at $x=(\mu ,0)$ and P₂ at $(-1+\mu ,0)$. (3) becomes,

$$\begin{eqnarray}&&\ddot{x}+2(-{\dot{x}}_{2},{\dot{x}}_{1})={\tilde{{\rm{\Omega }}}}_{x},\end{eqnarray} \tag{ 22 }$$

$\tilde{{\rm{\Omega }}}=(1/2)| x{| }^{2}+(1-\mu ){r}_{1}^{-1}+\mu {r}_{2}^{-1}+(1/2)\mu (1-\mu )$, ${r}_{1}=| (x-(\mu ,0)|$, ${r}_{2}=| x-(-1+\mu ,0)|$. The Jacobi integral function, $J(x,\dot{x})$ for this system is given by

$$\begin{eqnarray}&&J=2\tilde{{\rm{\Omega }}}-| \dot{x}{| }^{2}.\end{eqnarray} \tag{ 23 }$$

The differential equations have 5 well known equlibrium points, L_i, $i=1,2,3,4,5$, where ${L}_{1},{L}_{2},{L}_{3}$ are the collinear Lagrange points, and ${L}_{4},{L}_{5}$ are equilateral points. We assume the convention that L₂ lies between ${P}_{1},{P}_{2}$. $J{| }_{{L}_{i}}={C}_{i}$, where $3={C}_{4}={C}_{5}\lt {C}_{3}\lt {C}_{1}\lt {C}_{2}$. The collinear points are all local saddle-center points with eigenvalues, $\pm \alpha$ and $\pm i\beta$, $\alpha \gt 0,\beta \gt 0,{i}^{2}=-1$. The equilateral points ${L}_{4},{L}_{5}$ are locally elliptic points. We will focus on ${L}_{1},{L}_{2}$ in our analysis. As is described in [8, 18], the distance of ${L}_{1},{L}_{2}$ to P₂, ${r}_{{L}_{j}}$, $j=1,2$, is ${r}_{{L}_{j}}={ \mathcal O }({\mu }^{1/3})$. ${C}_{j}=3+| { \mathcal O }({\mu }^{2/3})| \gtrsim 3$ for $\mu \gtrsim 0$.

Projecting the three-dimensional Jacobi surface ${J}^{-1}(C)$ into physical $({x}_{1},{x}_{2})$-space, yields the Hill's regions, where P₀ is constrained to move. (see [8], figure 3.6) For C slightly greater than C₂, $C\gtrsim {C}_{2}$, the Hill's regions about ${P}_{1},{P}_{2}$, labeled ${H}_{1},{H}_{2}$, respectively, are not connected, so that P₀ cannot pass from one region to another. There is also a third Hill's region, H₃ that surrounds both ${P}_{1},{P}_{2}$ disconnected from ${H}_{1},{H}_{2}$, where P₀ can move about both primaries. When $C={C}_{2}$, ${H}_{1},{H}_{2}$ are connected at the single Lagrange point L₂ and P₀ still cannot pass between the primaries. When $C\lesssim {C}_{2}$, a small opening occurs between ${P}_{1},{P}_{2}$ near the L₂ location, we refer to as a neck region, N₂, first discussed in [19]. When C decreases further, $C\lesssim {C}_{1}$, another opening occurs near L₁ and forms another neck region, N₁, that connects H₂ with the outer Hill's region, H₃.

A retrograde unstable hyperbolic periodic orbit is contained in N₂, we label ${\gamma }_{2}$. ${\gamma }_{2}$ has local stable and unstable two- dimensional manifolds ${M}_{j}^{s}({\gamma }_{2}),{M}_{j}^{u}({\gamma }_{2})$, $j=1,2$, which extend from N₂ into H_j. These manifolds are topologically equivalent to two-dimensional cylinders. It is shown in [19] that orbits can only pass from H₂ to H₁, or from H₁ to H₂, by passing within the three-dimensional region contained inside ${M}_{j}^{s}({\gamma }_{2}),{M}_{j}^{u}({\gamma }_{2})$, which are called transit orbits. For example, to pass from H₂ to H₁, P₀ must pass into the three-dimensional region inside ${M}_{2}^{s}({\gamma }_{2})\subset {H}_{2}$ and out from the region inside ${M}_{1}^{u}({\gamma }_{2})\subset {H}_{1}$ (see figure 2) (see also [8], figure 3.9). N₂ is bounded on either side of P₂ by vertical lines ${l}_{R},{l}_{L}$, that cut the x₁-axis, to the right and left of P₂, respectively. On the Jacobi surface, $\{J=C\}$, ${J}^{-1}({N}_{2})$ is a set with topological two-dimensional spheres as boundaries, ${S}_{R}^{2},{S}_{L}^{2}$ corresponding to the lift of ${l}_{R},{l}_{L}$, respectively, onto $\{J=C\}$. When a transit oribit passes from H₂ to H₁, then on the Jacobi surface, P₀ passes from ${S}_{L}^{2}$ to ${S}_{R}^{2}$. The bounding spheres separate ${H}_{1},{H}_{2}$ from N₂.

Zoom In Zoom Out Reset image size

For $C\lesssim {C}_{1}$, N₁ contains the Lyapunov orbit ${\gamma }_{1}$. Manifolds, ${M}_{j}^{s}({\gamma }_{2}),{M}_{j}^{u}({\gamma }_{2})$, $j=2,3$, are similarly obtained where transit orbits can pass between H₂ and H₃, passing through the respective bounding spheres. The geometry in this case is shown in figure 3).

Zoom In Zoom Out Reset image size

It is noted that in center of mass, rotating coordinates, $({x}_{1},{x}_{2}$), (5) becomes,

$$\begin{eqnarray}&&{E}_{2R}=\displaystyle \frac{1}{2}| \dot{x}{| }^{2}+\displaystyle \frac{1}{2}| x{| }^{2}-\omega ({\dot{x}}_{1}{x}_{2}-{\dot{x}}_{2}{x}_{1})-\displaystyle \frac{G({m}_{0}+{m}_{2})}{{r}_{2}},\end{eqnarray} \tag{ 24 }$$

with $\omega =1,{m}_{0}=0,{m}_{2}=\mu ,G=1$.

Finally, a translation, ${z}_{1}={x}_{1}-(-1+\mu ),{z}_{2}={x}_{2}$, is made to a P₂-centered coordinate system, $({z}_{1},{z}_{2})$, where P₁ is at $(1,0)$, and ${r}_{2}=| z|$. For notation, we set $r\equiv {r}_{2}$. We refer to ${E}_{2R}$ in center of mass rotating coordinates $({x}_{1},{x}_{2})$ and also in P₂-centered rotating coordinates $({z}_{1},{z}_{1})$, where ${E}_{2R}$ is a different expression from (24).

The line ${ \mathcal L }$ emanates from P₂ and makes an angle ${\theta }_{2}\in [0,2\pi ]$ with respect to the z₁-axis. Trajectories of P₀ are propagated from ${ \mathcal L }$ such that at each point on ${ \mathcal L }$ at a distance $r\gt 0$, the eccentricity, e₂, is kept fixed to a value, ${e}_{2}\in [0,1)$, by adjusting the velocity magnitude, whose initial direction is perpendicular to ${ \mathcal L }$. Also, the velocity direction is assumed to be clockwise (similar results are obtained for counter clockwise propogation). The initial points of propagation on ${ \mathcal L }$ are periapsis points of an osculating ellipse of velocity ${v}_{p}=\sqrt{G({m}_{0}+{m}_{2})(1+{e}_{2})/r}-\omega r$, $\omega =1,{m}_{0}=0,{m}_{2}=\mu$. It is noted that ${ \mathcal L }$ makes a two-dimensional surface of section, defined in polar coodinates, ${S}_{{\theta }_{2}^{0}}=\{({r}_{2},{\theta }_{2},{\dot{r}}_{2},{\dot{\theta }}_{2}| {\theta }_{2}={\theta }_{2}^{0},{\dot{\theta }}_{2}\gt 0\}$. It is also noted that as r changes on ${ \mathcal L }$, the Jacobi energy also changes. This implies that ${W}_{k}^{u},{W}_{k}^{s},{ \mathcal W }$ do not lie on a fixed Jacobi surface. Also the Hill's regions vary within these sets.

As is described in [13, 17], a sequence of consecutive open intervals, ${I}_{j}^{k}$, are obtained along ${ \mathcal L }$, for a fixed ${\theta }_{2},{e}_{2}$, that alternate between stable and unstable points, for k cycles. That is, ${I}_{1}^{k}=\{{r}_{0}^{k}\lt r\lt {r}_{1}^{k}\}$, ${r}_{0}^{k}=0$, are stable points, ${I}_{2}^{k}=\{{r}_{1}^{k}\lt r\lt {r}_{2}^{k}\}$ are unstable points, etc (see figure 4) There are ${N}_{k}({\theta }_{2},{e}_{2})$ stable sets, and unstable sets, for an integer ${N}_{k}\geqslant 1$. The boundary points ${r}_{j}^{k},j=1,2,\,\ldots ,\,{N}_{k}({\theta }_{2},{e}_{2})$, represent the transition between stable and unstable points relative to k cycles, where the kth unstable points lead to stable motion for $k-1$ cycles and are unstable on the kth cycle. The kth stable set for a given value of ${\theta }_{2},{e}_{2}$ is given by,

$$\begin{eqnarray}&&{W}_{k}^{s}({\theta }_{2},{{\rm{e}}}_{2})=\bigcup _{j=0}^{{N}_{k}({\theta }_{2},{e}_{2})}({r}_{2j}^{k},{r}_{2j+1}^{k}).\end{eqnarray} \tag{ 25 }$$

This is a slice of the entire stable set, ${W}_{k}^{s}$, by varying ${\theta }_{2},{e}_{2}$, given by

$$\begin{eqnarray}&&{W}_{k}^{s}=\bigcup _{\theta \in [0,2\pi ],e\in [0,1)}{W}_{k}^{s}({\theta }_{2},{{\rm{e}}}_{2}).\end{eqnarray} \tag{ 26 }$$

Zoom In Zoom Out Reset image size

We define ${{ \mathcal W }}_{k}=\partial {W}_{k}^{s}$. ${{ \mathcal W }}_{k}$ has a Cantor-like structure as is described in [13]. The numerical estimation of ${W}_{k}^{s}$, ${{ \mathcal W }}_{k}$ is given in [13, 17, 20], for different values of k, μ, ${\theta }_{2}$, e₂. The motion of P₀ is seen to be unstable and sensitive for initial conditions near ${{ \mathcal W }}_{k}$. It is remarked that due to limitations of computer processing time, k is not taken too large.

A main result of [13], is that ${{ \mathcal W }}_{k}$ about P₂ is equivalent to the set of global stable manifolds, ${M}_{2}^{s}({\gamma }_{1})\cup {M}_{2}^{s}({\gamma }_{2})$, to the Lyapunov orbits, ${\gamma }_{1},{\gamma }_{2}$, respectively, about the collinear Lagrange points, ${L}_{1},{L}_{2}$, on either side of P₂ for $C\lesssim {C}_{1}$ and μ sufficiently small, in H₂. Similarly, one could restrict $C\lesssim {C}_{2}$ and have equivalence to only the global manifold ${M}_{2}^{s}({\gamma }_{2})$ in H₂. This is demonstrated numerically by examining the intersections of ${M}_{2}^{s}({\gamma }_{1}),{M}_{2}^{s}({\gamma }_{2})$ on surfaces of section ${S}_{{\theta }_{2}^{0}}$ satisfying $\dot{r}=0,{E}_{2}\lt 0$ for μ sufficiently small, and varying $0\leqslant {\theta }_{2}^{0}\leqslant 2\pi$. It is shown in [13] that a very small set of points exist on ${{ \mathcal W }}_{k}$ that do not satisfy this equivalence. These points are not considered⁵ .

The reason this equivalence is true is due to the separatrix property of the manifolds (see [13, 19]). Assume $C\lesssim {C}_{2}$. The separatrix property means that if a trajectory point is inside of the region bounded by ${M}_{2}^{s}({\gamma }_{2})$ on ${ \mathcal L }$, it will wind about P₂ staying inside the region contained by ${M}_{2}^{s}({\gamma }_{2})$ as ${M}_{2}^{s}({\gamma }_{2})$ winds about P₂. P₀ can't go outside this manifold region. Eventually, ${M}_{2}^{s}({\gamma }_{2})$ will go to ${\gamma }_{2}$, and P₀ will pass through N₂ into H₁ as a transit orbit, after it makes $k-1$ complete cycles, before completing the kth cycle. This corresponds to an unstable point on the set ${W}_{k}^{u}$. If a trajectory point is outside ${M}_{2}^{s}({\gamma }_{2})$ on ${ \mathcal L }$, P₀ will remain in H₂, making k complete cycles about P₂ near the outside of ${M}_{2}^{s}({\gamma }_{2})$, but it can't escape to H₁. Thus, ${M}_{2}^{s}({\gamma }_{2})$ itself is equivalent to ${{ \mathcal W }}_{k}$. That is, ${M}_{2}^{s}({\gamma }_{2})$ separates between stable and unstable motion.

The intersections of ${M}_{2}^{s}({\gamma }_{2})$ on ${ \mathcal L }$ in physical space as it cycles around P₂ give rise to the alternating intervals between stable and unstable motion, $\{{I}_{1}^{k},{I}_{2}^{k},\ldots \}$, where there are ${N}_{k}({\theta }_{2},{e}_{2})$ such intervals. Points inside ${M}_{2}^{s}({\gamma }_{2})$ on ${ \mathcal L }$ correspond to points in the set ${W}_{k}^{u}$, and points outside of ${M}_{2}^{s}({\gamma }_{2})$ on ${ \mathcal L }$, and close to it, correspond to points of the set ${W}_{k}^{s}$.

The relationship between the manifolds and ${W}_{k}^{s},{W}_{k}^{u}$ is shown in figure 5.

Zoom In Zoom Out Reset image size

It can be shown that if ${M}_{2}^{s}({\gamma }_{2})$ has transverse intersections, which is numerically demonstrated, where the manifold tube breaks, the separatrix property is still satisfied even though a section through the tube no longer gives a circle, but rather parts of broken up circles.

Numerical simulations in [13, 18], indicate for $C\lesssim {C}_{1}$, that ${M}_{2}^{s}({\gamma }_{1})$ can intersect ${M}_{2}^{u}({\gamma }_{2})$ transversally, giving rise to a complex network of invariant manifolds about P₂, and for $C\lesssim {C}_{2}$, ${M}_{2}^{u}({\gamma }_{2})$ can intersect ${M}_{2}^{s}({\gamma }_{2})$ transversally, for a set of μ and C. This supports the fact that the motion near ${{ \mathcal W }}_{k}$ is sensitive.

Let $\zeta (t)=({z}_{1}(t),{z}_{2}(t),{\dot{z}}_{1}(t),{\dot{z}}_{2}(t))\in {{\mathbb{R}}}^{4}$ be the trajectory of P₀ in rotating P₂-centered coordinates, and $z(t)=({z}_{1}(t),{z}_{2}(t))\in {{\mathbb{R}}}^{2}$ the trajectory of P₀ in physical coordinates. Similarly, in inertial P₂ centered coordinates, we define, ${ \mathcal Z }(t)=({Z}_{1}(t),{Z}_{2}(t),{\dot{Z}}_{1}(t),{\dot{Z}}_{2}(t)$, and $Z(t)=({Z}_{1}(t),{Z}_{2}(t))$. We will use inertial and rotating coordinates to describe the motion of P₀,

The following result, referenced previously, is proven,

Proposition 3.1 Assume ${P}_{0}\in {W}_{k}^{u}$ at time $t={t}_{0}$, which implies ${E}_{2}({t}_{0})\lt 0$. There are two possibilities: (i.) P₀ cycles about P₂ $k-1$ times, then moves to cycle about P₁ without cycling about P₂. This implies that P₀ weakly escapes P₂. That is, there exists a time ${t}^{* }\gt {t}_{0}$, after the $(k-1)$ st cycle where ${E}_{2}({t}^{* })=0$, ${E}_{2}\gtrsim 0$ for $t\gtrsim {t}^{* }$, and ${E}_{2}(t)\lt 0$ for ${t}_{0}\leqslant t\lt {t}^{* }$, (ii.) P₀ does k complete cycles about P₂, where on the kth cycle P₀ returns to ${ \mathcal L }$ with ${E}_{2}\gt 0$. (It is assumed the set of collision orbits to P₁ and P₂, Γ, are excluded which are a set of measure 0.)

Proof of proposition 3.1–(i.). This is shown to be true by noting that when P₀ does a cycle about P₁, it will cross the x₁-axis, where ${r}_{2}\gt 1+c$, $c\gt 0$. (24) implies that ${E}_{2R}=(1/2){\dot{x}}_{1}^{2}+(1/2){\mu }^{2}+{\dot{x}}_{2}(1+c)-\mu {\left(1+c\right)}^{-1}$, where $\dot{{x}_{2}}\gt 0$, $\mu \ll 1$. This implies there exists a time ${t}^{* * }$ where ${E}_{2R}\gt 0$. Since ${E}_{2R}\lt 0$ at $t={t}_{0}$, then there exists a time ${t}_{0}\lt {t}^{* }\lt {t}^{* * }$ where ${E}_{2R}=0$ and ${E}_{2}\gtrsim 0$ for $t\gtrsim {t}^{* }$. (ii.) This yields weak capture since ${E}_{2}\lt 0$ for $t={t}_{0}$ and E₂ becomes positive. Thus, there exists a time ${t}^{* }$ where ${E}_{2}({t}^{* })=0$, then becomes slightly positive.

Global trajectory after weak capture

We rigorously prove result A, that after weak capture with respect to P₂, P₀ can move onto a resonance orbit about P₁ in resonance with P₂, and then return to weak capture. This is done by a series of Propositions.

The following sets are defined for trajectories for P₀ starting in weak capture at $t={t}_{0}$ that go to weak escape at a time ${t}_{1}\gt {t}_{0}$.

Assumptions A. 

• Type I = {$\zeta =(z,\dot{z})$ at t₀ is on or near ${W}_{k}^{u}$ (${E}_{2}\lt 0$, $\dot{r}=0$ or $| \dot{r}| \gtrsim 0$, resp.)}
• Type II = {$\zeta ({t}_{0})=(z({t}_{0}),\dot{z}({t}_{0}))$ is not near ${W}_{k}^{u}$, where $| \dot{r}({t}_{0})|$ is not near 0 and ${E}_{2}({t}_{0})\lt 0$ }
• Type IIa = {$\zeta ({t}_{0})$ is a Type II point where $\zeta (t)$ goes to weak escape at ${t}_{1}\gt {t}_{0}$ with $| \dot{r}({t}_{1})|$ not near 0 } (i.e. there is no cycling about P₂.)
• Type IIb = {$\zeta ({t}_{0})$ is a Type II point where there exists a time $\hat{t}\gt {t}_{0}$, such that $\zeta (\hat{t})$ on or near ${W}_{k^{\prime} }^{u}$, for some integer $k^{\prime} \geqslant 1$ }
• Case A = {${P}_{0}$ cycles about ${P}_{2}(k-1)$ times, then moves to cycle about ${P}_{1}$ }
• Case B = { ${P}_{0}$ does not cycle about ${P}_{1}$ after $(k-1)$ st cycle. Instead, on $k$-th cycle about ${P}_{2},{P}_{0}$ returns to ${ \mathcal L }$ with ${E}_{2}\gt 0$ }
• Γ = { ${P}_{0}$ goes to collision with ${P}_{1}$ or ${P}_{2}$ for $t\gt {t}_{0}$ }

Result A is stated more precisely as,

Theorem A. Assume P₀ is weakly captured at a distance r from P₂ at $t={t}_{0}$. Assume the weak capture point, $\zeta ({t}_{0})$ is of Type I, Type IIb, Case A, which are numerically observed to be generic [13], and assume the following sets are ruled out: Type IIa, Case B, Gamma (numerically observed to be small [13]). Assume also that $C\lesssim {C}_{2}$, μ sufficintly small. Then P₀ will escape P₂ through N₂ by passing within the region contained within ${M}_{2}^{s}({\gamma }_{2})\subset {H}_{2}$, and moving into the H₁ through the region within ${M}_{1}^{u}({\gamma }_{2})$. This escape is approximately parabolic since ${E}_{2}\approx 0$ on ${S}_{R}^{2}\subset {H}_{1}$. (Parabolic escape is when there exists a time $t\gt {t}_{0}$ where ${E}_{2}=0$.) $\zeta (t)$ evolves into an approximate resonance orbit about P₁ with an apoapsis near ${S}_{R}^{2}$ of N₂, peforming several cycles about P₁, then returns to ${S}_{R}^{2}$ passing through N₂ within the region contained within ${M}_{1}^{s}({\gamma }_{2})$ and exiting N₂ through the interior of ${M}_{2}^{u}({\gamma }_{2})$ and moving onto weak capture about P₂. The process repeats unless P₀ moves on any of the sets: Type IIa, Case B, Γ, or P₀ escapes the ${P}_{1},{P}_{2}$-system.

If $C\lesssim {C}_{1}$, then P₀ can parabolically escape P₂ through ${S}_{R}^{2}$ of N₂, as previously described obtaining a sequence of resonance orbits in H₁, or it can parabolically escape P₂ through ${S}_{L}^{2}=\partial {N}_{1}$ into H₃, by passing through the region within ${M}_{2}^{s}({\gamma }_{1})$ and exiting from the region within ${M}_{3}^{u}({\gamma }_{1})$, and form a larger resonance orbit about P₁ with a periapsis near ${S}_{L}^{2}$ in H₃, which eventually returns to weak capture about P₂ though N₁, reversing the previous pathway. This process terminates if P₀ moves on any of the sets Type IIa, Case B, Γ or escapes the ${P}_{1},{P}_{2}$ system.

This yields a sequence of approximate resonance orbits depending on the choice of the weak capture initial condition. The set of all such resonance orbits form the family, ${\mathfrak{F}}$. The frequencies of these orbits satisfy (6) of lemma A.

Proposition 3.2 Let P₀ be captured with respect to P₂ at a distance r from P₂ at a time t₀, where ${E}_{2}({ \mathcal Z }({t}_{0}))={E}_{2R}(\zeta ({t}_{0}))\lt 0$. Then, P₀ is weakly captured at t₀. That is, P₀ moves to weak escape at a time $t={t}^{* }\gt {t}_{0}$, where ${E}_{2}({ \mathcal Z }({t}^{* }))=0,\hspace{.05in}{E}_{2}({ \mathcal Z }(t))\gtrsim 0,t\gtrsim {t}^{* }$. (It is assumed Type IIa, Case B, Γ points are excluded.)

Proof of proposition 3.2. We distinguish several types of weak capture points.

Type I is where P₀ is on or near ${W}_{k}^{u}$ at $t={t}_{0}$. In this case, P₀ is at a distance r from P₂ where $\dot{r}=0$ or $| \dot{r}| \gtrsim 0$, where we have made use of the fact ${W}_{k}^{u}$ is open, so that ${e}_{2}\lt 1$ and ${E}_{2}({t}_{0})\lt 0$. Thus, P₀ is captured at $t={t}_{0}$. For $t\gt {t}_{0}$, the proof follows by proposition 3.1.

Type II is where P₀ is not near ${W}_{k}^{u}$ since $| \dot{r}|$ is not near 0 at $t={t}_{0}$. There are two types. Type IIa is where P₀ starts at $t={t}_{0}$ with E₂ and then to weak escape, with no cycling, by definition. If P₀ starts on a Type IIb point for $t={t}_{0}$, then for $t\gt {t}_{0}$ there will be a time $\hat{t}\gt {t}_{0}$ where $\dot{r}=0$ or $| \dot{r}| \gtrsim 0$. In that case, P₀ is on or near ${W}_{k^{\prime} }^{u}$ at $t=\hat{t}$, for some $k^{\prime} \geqslant 1$. This yields a Type I point, that implies weak capture.

In all these cases, P₀ moves to weak escape at a time $t={t}^{* }\gt {t}_{0}$, where ${E}_{2R}(\zeta ({t}^{* }))=0$, ${E}_{2R}(\zeta (t))\gtrsim 0,t\gtrsim {t}^{* }$. This proves proposition 3.2.

As in [13], we exclude Type IIa points as they are not generic. Points on Γ are a set of measure 0 and can be omitted. Case B points are non-generic and excluded.

We now determine what kind of motion P₀ has about P₂ for times up to weak escape at ${t}^{* }$. Consider the trajectory of P₀ as it undergoes counterclockwise cycling about P₂ after leaving points on or near ${W}_{k}^{u}$ on a line ${ \mathcal L }$ in both Types I, IIb. (similar results are obtained for clockwise cycling) As P₀ performs $k-1$ cycles, it either has weak escape prior to completing the kth cycle, where ${E}_{2}=0$, and then when it intersects ${ \mathcal L }$, ${E}_{2}\gt 0$, we call Case B, or it moves to cycle P₁ after the $(k-1)$ st cycle where it was shown in proposition 3.1 that P₀ achieves weak escape, we refer to as Case A.

Proposition 3.3 Assume ${P}_{0}\in {W}_{k}^{u}$ at t=0, $C\lesssim {C}_{2}$, assuming Case A, and excluding Case B. Then after $(k-1)$-cycles about P₂, P₀ moves away from P₂, passing through the interior region of ${M}_{2}^{s}({\gamma }_{2})$ into N₂, between H₂ and H₁, through the interior of ${M}_{1}^{u}({\gamma }_{2})$, into H₁ where it starts to cycle P₁. When P₀ is within N₂, ${E}_{2}\lesssim 0$. If $C\lesssim {C}_{1}$, then after $(k-1)$-cycles about P₂, P₀ moves away from P₂, passing through the interior of ${M}_{2}^{s}({\gamma }_{2})$ into N₂ between H₂ and H₁, and out into H₁ as before, or P₀ passes through the interior of ${M}_{2}^{s}({\gamma }_{1})$ into N₁ between H₂ and H₃, and out into H₃ through the interior of ${M}_{3}^{u}({\gamma }_{1})$. When P₀ is within N₁, ${E}_{2}\lesssim 0$.

Proof of proposition 3.3. Case A is considered with $C\lesssim {C}_{2}$. P₀ starts on ${ \mathcal L }$ at $t={t}_{0}$ with ${E}_{2}\lt 0$, $\dot{r}=0$. It cycles about P₂, completes the $(k-1)$ st cycle, then moves to H₁ where it starts to cycle about P₁, where ${\dot{\theta }}_{1}\gt 0$, for $0\leqslant {\theta }_{1}\leqslant 2\pi$ (see [8]). By the separatrix property P₀ is within the interior region contained by ${M}_{2}^{s}({\theta }_{2})$ on ${ \mathcal L }$ at $t={t}_{0}$ and it must pass from P₂, through N₂, where it is a transit orbit [13]. When P₀ passes through N₂ it must pass inside the region bounded by ${M}_{2}^{s}({\gamma }_{2})$, and emerge from N₂ inside the region bounded by ${M}_{1}^{u}({\gamma }_{2})$ at ${S}_{R}^{2}$, where it will begin to cycle about P₁. For μ sufficiently small, the width of N₂ is near 0, and geometrically this implies the velocity of P₀ with respect to P₂ is near 0 since it passes close to L₂ in phase space.

When ${P}_{0}\in {S}_{R}^{2}$ in H₁, the distance from P₀ to P₂ can be estimated. The value of $C\lesssim {C}_{2}$, and ${C}_{2}=3+{\left(\mu /3\right)}^{2/3}+{ \mathcal O }(\mu /3)$. P₂ is near L₂. It directly follows that $r\equiv {r}_{2}={\left(\mu /3\right)}^{1/3}+\left.{ \mathcal O }\left({\left(\mu /3\right)}^{2/3}\right)\right)$. (This implies, ${r}_{1}=1-| { \mathcal O }\left({\left(\mu /3\right)}^{1/3}\right)$ ) $| \lesssim 1$ since P₀ is slightly to the right of L₂ at ${S}_{R}^{2}$.)

The estimate of r₂ implies that for μ sufficiently small ${r}_{2}\approx \mu /3$. Also, at L₂, $\dot{x}=0$. Thus, equation (24) implies

$$\begin{eqnarray}&&{E}_{2R}=(-{3}^{1/3}+(1/2){3}^{-2/3}){\mu }^{-2/3}+{ \mathcal O }({\mu }^{b})\lesssim 0,\end{eqnarray} \tag{ 27 }$$

$b\gt 2/3$.

Case A is considered with $C\lesssim {C}_{1}$. (As r increases along ${ \mathcal L }$, keeping a constant eccentricity, C will decrease and move slightly below the other value, C₁ for L₁, 180 degrees away from L₂ on the anti-P₁ side of P₂, $C\lesssim {C}_{1}$.) As t increases from t₀, by the separatrix property, P₀ has two possibilities: (i) P₀ can pass through the region bounded by ${M}_{2}^{s}({\gamma }_{1})$, through N₁ and exiting within the region bounded by ${M}_{3}^{u}({\gamma }_{1})$ into H₃ intersecting ${S}_{L}^{2}=\partial {N}_{1}$ at a time t₁. It can then start to cycle about P₁ in H₃ for $t\gt {t}_{1}$, where ${\dot{{theta}}}_{1}\gt 0$. This implies unstable motion occurs, where after $(k-1)$-cycles about P₂, P₀ starts to cycle about P₁ in the H₃ region. It is similarly verified that (27) is satisfied on ${S}_{L}^{2}$ at $t={t}_{1}$. This is different from when P₀ cycles about P₁ after $(k-1)$-cycles as it emerges from N₂ into the H₁ region. However, in both cases, as seen, ${E}_{2}\lesssim 0$ in the neck region bounding spheres. (ii) P₀ passes through N₂ into H₁. This yields the same results as in Case A. (It is verified that $C\lesssim {C}_{1}$ is sufficient to yield the same estimates in this proof for ${E}_{2R}$ as obtained for $C\lesssim {C}_{2}$.)

In summary, given ${P}_{0}\in {W}_{k}^{u}$ at time $t={t}_{0}$ there exists a time ${t}_{1}\gt {t}_{0}$ where ${E}_{2}\lesssim 0$ which occurs at ${S}_{R}^{2}\subset {H}_{1}$ for $C\lesssim {C}_{2}$, and for $C\lesssim {C}_{1}$, ${E}_{2}\lesssim 0$ on ${S}_{L}^{2}\subset {H}_{3}$, or on ${S}_{R}^{2}\subset {H}_{1}$. Thus, in both cases, approximate parabolic escape occurs.

In the next step, we see what happens as P₀ starts to move about P₁ after leaving ${S}_{R}^{2}=\partial {N}_{2}$ in H₁, or ${S}_{L}^{2}=\partial {N}_{1}$ in H₃, through ${M}_{1}^{u}({\gamma }_{2})$, or ${M}_{3}^{u}({\gamma }_{1})$, respectively.

Proposition 3.4 Assume ${P}_{0}\in {S}_{R}^{2}\subset {H}_{1}$ at $t={t}_{1}$. P₀ moves from ${S}_{R}^{2}$ for $t\gt {t}_{1}$ into an approximate resonance orbit about P₁. After j cycles, $j\geqslant 1$, P₀ returns to ${S}_{R}^{2}$ where ${E}_{2}\lesssim 0$. It then moves through N₂ to weak capture by P₂.

(Similarly, assuming ${P}_{0}\in {S}_{L}^{2}\subset {H}_{3}$ at $t={t}_{1}$, P₀ moves from ${S}_{L}^{2}$ for $t\gt {t}_{1}$ into an approximate resonance orbit about P₁ in the outer Hills region H₃. After j cycles, $j\geqslant 1$, about P₁, P₀ returns ${S}_{L}^{2}$ where it then moves through N₁ to weak capture by P₂.)

Proof of proposition 3.4. The case of ${P}_{0}\in {S}_{R}^{2}\subset {H}_{1}$ is considered first, where $t={t}_{1}$, $C\lesssim {C}_{2}$. [21] is referenced since it determines the set ${{ \mathcal W }}_{k}$ about the larger primary P₁ analytically.

When ${P}_{0}\in {S}_{R}^{2}$ for $t={t}_{1}$, this implies it lies in the three-dimensional region bounded by ${M}_{1}^{u}$. Moreover, for $t\gt {t}_{1}$, due to the separatrix property, P₀ stays within this region inside ${M}_{1}^{u}$ for all time moving forward [21]. This manifold stays within a bounded region, ${{\mathfrak{M}}}_{1}$, bounded by the following: ${S}_{R}^{2}$, the boundary of H₁ (a zero velocity curve), and a two-dimensional McGehee torus, T_M, about P₁ [18, 21] ⁶ . The width of ${{\mathfrak{M}}}_{1}$ is ${ \mathcal O }({\mu }^{1/3})$.

There are two cases. The first is where ${M}_{1}^{u}$ is a homoclinic two-dimensional tube which transitions from ${M}_{1}^{u}$ to ${M}_{1}^{s}$ which goes to ${S}_{R}^{2}$. This implies P₀ returns to ${S}_{R}^{2}$ at a later time. Now, if ${M}_{1}^{u}$ intersects ${M}_{1}^{s}$ transversally, then these manifolds intersect in a complex manner, where the image of ${M}_{1}^{u}$ on two-dimensional sections, ${S}_{{\theta }_{1}^{0}}$, are not circles, but parts of circles after several cycles of P₀ about P₁, However, the separatrix property is still preserved, and P₀ still returns to ${S}_{R}^{2}$ [21].

Let's assume it returns to ${S}_{R}^{2}$ after a time T, (${t}_{2}={t}_{1}+T$). P₀ is a transit orbit and must pass through N₂ for $t\gt {t}_{2}$ into H₂ through the interior region bounded by ${M}_{2}^{u}$, where it is again weakly captured by P₂. This follows since when P₀ passes through N₂ into H₂, within the interior region bounded by ${M}_{2}^{u}$, it will intersect ${S}_{L}^{2}\subset {N}_{2}$ in H₂. The estimate obtained in (27) is also obtained at ${S}_{L}^{2}$. This implies P₀ is captured by P₂ at ${S}_{L}^{2}\subset {N}_{2}$ at a time ${t}_{2}+\delta$, $\delta \gt 0$. Under the previous assumptions on capture points in theorem A, P₀ is weakly captured and weakly escapes P₂.

The motion of P₀ as it leaves weak capture near P₂, passing into the H₁ region and moving back to the H₂ to weak capture is illustrated in figure 6.

Zoom In Zoom Out Reset image size

It is noted that there exists a time ${\tilde{t}}_{3}\lt {t}_{2}$ where ${E}_{2}\gt 0$, which follows from the proof of proposition 3.1. Thus, P₀ is weakly captured in backwards time at $t={t}_{2}+\delta$.

A similar argument holds for $C\lesssim {C}_{1}$. Within H₂ there are openings at N₁ to the left of P₂ and N₂ to the right. P₀ can now move into H₃ through N₁, in addition to moving into H₁ through N₂, from weak capture points on ${W}_{k}^{u}\cap { \mathcal L }$ in H₂ after j cycles. If P₀ moves into H₁, it does so from the region bounded by ${M}_{1}^{u}({\gamma }_{2})$ and the same argument follows from the case $C\lesssim {C}_{2}$.

If P₀ moves about P₁ in H₃, it moves in a bounded region ${{\mathfrak{M}}}_{3}$. This region is bounded by ${S}_{L}^{2}=\partial {N}_{1}$ in H₃, a McGehee torus T_M about P₁ in H₃ and the boundary of H₃. P₀ moves inside the region enclosed by ${M}_{3}^{u}({\gamma }_{1})$ and stays within it as this manifold either transitions into ${M}_{3}^{s}({\gamma }_{1})$ as a homoclinic tube or if the manifolds have transverse intersection. The separatrix property is satisfied, and P₀ will cycle about P₁ in H₃ j times until transits into H₂ and intersects ${S}_{R}^{2}=\partial {N}_{1}$ in H₂, where (27) is satisfied which implies P₀ is captured by P₂ at a time ${t}_{2}+\delta$. Under the assumptions of theorem A, P₀ is weakly captured at P₂.

It is noted P₀ is weakly captured in backwards time, since there is a time ${t}_{3}\lt {t}_{2}$ where ${E}_{2}\gt 0$, when P₀ was moving in H₃, using the same argument as in the proof of proposition 3.1.

The final part of the proof of proposition 3.4 is to prove that P₀ moves in resonance orbits about P₁.

We consider $C\lesssim {C}_{2}$, where P₀ is moving in H₁. P₀ is on an approximate resonance orbit in H₁ about P₁ for ${t}_{1}\leqslant t\leqslant T$. This is proven as follows: P₀ moves in ${{\mathfrak{M}}}_{1}$. The orbit for P₀ will not deviate too much for μ sufficiently small, by the amount ${ \mathcal O }({\mu }^{1/3})$ [18]. It is an approximate elliptic Keplarian orbit about P₁, since its energy ${E}_{1}\lt 0$(proven in the following text, see proposition A). It has a uniform approximate Keplerian period, T₁, for μ sufficiently small, with approximate frequency ${\omega }_{1}={T}_{1}^{-1}$. Once P₀ moves away from ${S}_{R}^{2}$ for $t\gt {t}_{1}$, and returns to ${S}_{R}^{2}$ for $t={t}_{2}=T+{t}_{1}$. When P₀ returns to ${S}_{R}^{2}$, it returns to near P₂ to approximately the distance ${ \mathcal O }\left({\left(\mu /3\right)}^{2/3}\right)$, as follows from the proof of proposition 3.3.

Since P₀ returns to ${S}_{R}^{2}$, near to P₂, T must approximately be an integer multiple, n, of the period, T₂, of P₂ about P₁. That is, $T\approx {{nT}}_{2}$. Also, since P₀ returns to near where it started, $T\approx {{mT}}_{1}$. Thus, ${{mT}}_{1}\approx {{nT}}_{2}$. Equivalently, $n{\omega }_{1}\approx m\omega$. Thus, P₀ moves in an approximate $n:m$ resonance with P₂. It is noted that the approximate elliptic orbits of P₀ have an apoapsis distance from P₁ that is approximately the distance of ${S}_{R}^{2}$ to P₁.

This can also be visualized in inertial coordinates, $({Y}_{1},{Y}_{2})$ centered at P₁. When P₀ has started its motion on a near ellipse, for $t\gt {t}_{1}$, it has just left weak capture from near P₂ at the location, ${Y}^{* }$. P₀ then cycles about P₁ and keeps returning to near ${Y}^{* }$ each approximate period T₁. When it arrives near ${Y}^{* }$, P₂ needs to be nearby as when P₀ started its motion. Otherwise, P₀ won't become weakly captured by P₂ and leave the ellipse to move to the H₂ to weak capture by P₂. In that case it will continue cycling about P₁. If it does return to near ${Y}^{* }$ and P₂ has also returned near to where it started also near ${Y}^{* }$, then this means P₂ has gone around P₁ approximately n times and P₀ has gone around P₁ approximately m times.

In the case where P₀ moves in the H₃ region after leaving ${S}_{L}^{2}$ for $t\gt {t}_{1}$, one also obtains a resonance orbit by an analogous argument. In this case, P₀ has a periapsis near ${S}_{L}^{2}=\partial {N}_{1}$ with respect to P₁, where ${S}_{L}^{2}=\partial {N}_{1}$ is near P₂ at a distance of approximately, ${ \mathcal O }({\left(\mu /3\right)}^{1/3})$. These resonance orbits in H₃ are much larger than the resonance orbits in H₁ since they move about both ${P}_{1},{P}_{2}$.

Proposition A.  ${E}_{1}\lt 0$ when P₀ moves in H₁ about P₁ in a resonance orbit.

Proof of propisition A. When P₀ moves for $t\gt {t}_{1}$ it moves in an approximate two-body manner for finite time spans, where the osculating eccentricty e₁ and semi-major axis a₁ vary only for a small amount amount by ${ \mathcal O }({\mu }^{1/3})$ since P₀ moves within ${{\mathfrak{M}}}_{1}$. The energy E₁ is estimated(in an inertial frame). Since at $t={t}_{1}$, ${V}_{1}({t}_{1})\approx 1-| z({t}_{1})|$. We can estimate $| z({t}_{1})|$ as roughly the distance of L₂ to P₂. This implies, $| z({t}_{1})| \approx {\alpha }^{1/3}+{ \mathcal O }({\alpha }^{2/3})$, $\alpha =\mu /3$, and ${r}_{1}\approx 1-{\alpha }^{1/3}+{ \mathcal O }({\alpha }^{2/3})$. Thus,

$$\begin{eqnarray}&&{E}_{1}\approx (1/2){\left(1-{\alpha }^{1/3}+{ \mathcal O }({\alpha }^{2/3}\right)}^{2}-(1-\mu ){\left(1-{\alpha }^{1/3}+{ \mathcal O }({\alpha }^{2}/3\right)}^{-1}.\end{eqnarray} \tag{ 28 }$$

Thus ${E}_{1}\approx -(1/2)+{ \mathcal O }({\alpha }^{1/3})$. This implies that ${E}_{1}\lt 0$ for μ sufficiently small. P₀ will then be moving on an approximate ellipse about P₁ of an eccentricity, ${e}_{1}\lt 1$. The apoapsis of this ellipse will approximately be ${r}_{a}\approx {r}_{1}({t}_{1})\approx 1-{\left(\mu /3\right)}^{1/3}$. The semi-major axis of the ellipse at $t={t}_{1}$ is approximately, ${a}_{1}\approx -(1-\mu )/(2{E}_{1})\approx (1-\mu )+{ \mathcal O }({\left(\mu /3\right)}^{1/3})$. ${e}_{1}\approx 1-({r}_{a}/{a}_{1})\lt 1$.

When P₀ moves in resonance orbits in H₃ for $C\lesssim {C}_{1}$, similar estimates are made where ${E}_{1}\lt 0$.

End of proof of proposition A.

The resonance orbits of P₀ about P₁ move in an approximate two-body fashion where the perturbation due to P₂ is negligible for finite time spans. Thus, ${\omega }_{1}(t)\approx (m/n)\omega$ until it enters either neck to move to weak capture near P₂.

We assume $C\lesssim {C}_{2}$ and examine what happens to the motion of P₀ near ${Y}^{* }$ when in $n:m$ resonance with P₂. P₀ is at a minimal distance to P₂ when near ${Y}^{* }$. In the rotating system, it is near ${S}_{R}^{2}$, and lies in the three-dimensional region contained within ${M}_{1}^{s}$, since it has been moving about P₁ within this region by the seperatrix property. At this minimal distance, ${M}_{1}^{s}$ is close enough to ${\gamma }_{2}$ so that it can connect with it, and P₀ can move as a transit orbit and move through N₂ and exit into H₁ through the three-dimensional interior region bounded by ${M}_{1}^{u}$. It is then captured by P₂, with ${E}_{2}\lesssim 0$ at ${S}_{L}^{2}$. An analogous arument holds when $C\lesssim {C}_{1}$.

Assuming the generic assumptions are satisfied for theorem A, P₀ will weakly escape P₂ and again move into H₁, or H₃, obtaining resonance orbits, satisfying, ${\omega }_{1}\approx (m^{\prime} /n^{\prime} )\omega$, for integers $n^{\prime} \geqslant 1,m^{\prime} \geqslant 1$. The set of all such resonance orbits forms the family ${\mathfrak{F}}$. This concludes the proof of theorem A.

An example of the geometry of resonance transitions for an observed comet, Oterma, from a $2:3$ to a $3:2$, in 1936, and then back from a $3:2$ to a $2:3$, in 1962, ([22, 23] ) is illustrated in figure 7. (See section 3.1 at the end of this section.)

Zoom In Zoom Out Reset image size

It is noted that the estimates of ${E}_{1},{E}_{2}$ in the proof of theorem 2 while P₀ moves in H_i, $i=1,2,3$, are observed in the motions of the resonating comets studied in [22]. It can be seen in [22] that when the comet Gehrels 3 was weakly captured by Jupiter(P₂) from a 2:3 resonance orbit into an approximate 3:2 resonance orbit, ${E}_{2}\lesssim 0$. Also, when the comet moved about the Sun(P₁) in an approximate 2:3 resonance orbit, ${E}_{2}\gt 0$ and ${E}_{1}\lt 0$.

For each resonance orbit obtained from the choice of the weak capture initial condition, (6) is satisfied, proving lemma A.

3.1. Examples of resonance orbits in ${\mathfrak{F}}$ and ${\mathfrak{U}}$.

In this section some of the results are expanded upon in section 2.

The family ${\mathfrak{F}}$ of resonance periodic orbits, ${{\rm{\Phi }}}_{m/n}$, are modeled in the plane by the restricted three-body problem. The planar modeling is justified in section 3. To try and model ${\mathfrak{F}}$ with quantum mechanical ideas, we therefore use planar modeling. Thus, we consider the planar, time independent, modified Schrödinger equation given in the Introduction by (1), obtained from the classical Schrödinger equation, (?), by replacing $\hslash$ by σ, and the potential is given by the three-body potential $\bar{V}$ derived from the planar restricted three-body problem. This partial differential equation is time independent.

The motivation of replacing $\hslash$ by σ is given by analogy A in section 2, where σ given by (11). The potential $\bar{V}$ is given by (15), obtained from $\hat{V}={V}_{1}+{V}_{2}$. It is recalled that V₁ is the potential due to P₁ and V₂ is the potential due to P₂, in an inertial P₁-centered coordinate system, $({Y}_{1},{Y}_{2})$. It is also recalled that P₂ moves about P₁ on the circular orbit, $\gamma (t)$, ${m}_{0}\gtrsim 0$, and as described in section 2 for Results A, it is necessary that ${m}_{0},{m}_{2}$ are sufficiently small for the approximations in assumptions 1.

As described in section 2 the modified Schrödinger equation is given by (16), where $\bar{V}$ is the average of $\hat{V}$, obtained by averaging V₂ over a cycle of P₂ about P₁ on γ, given by (14). For reference, we recall (16),

$$\begin{eqnarray}&&-\displaystyle \frac{{\sigma }^{2}}{2\nu }{{\rm{\nabla }}}^{2}{\rm{\Psi }}+\bar{V}{\rm{\Psi }}=E{\rm{\Psi }},\end{eqnarray} \tag{ 29 }$$

In the macroscopic scale for the masses, and relative distances, two dimensions is required, where, in the inertial P₁-centered coordinates, $({Y}_{1},{Y}_{2})$ ${{\rm{\nabla }}}^{2}\equiv \tfrac{{\partial }^{2}}{\partial {Y}_{1}^{2}}+\tfrac{{\partial }^{2}}{\partial {Y}_{2}^{2}}$. When solving (29), the three-dimensional problem is solved for generality. The three-dimensional problem is discussed when considering the quantum scale.

It is noted that the units of σ are ${m}^{2}{{kgs}}^{-4/3}$. This needs to match the units of $\hslash$ which are ${m}^{2}{{kgs}}^{-1}$.($\hslash =6.626\,07\times {10}^{-34}{m}^{2}{{kgs}}^{-1}$) Thus, we need to multiply σ by $\rho =1{s}^{1/3}$, ${\sigma }^{* }=\rho \sigma$. ${\sigma }^{* }$ has the same units as $\hslash$. Keeping the same notation, ${\sigma }^{* }\equiv \sigma$. At the end of this section it is seen that when the masses approach the quantum scale, σ also gets small, and ${m}_{0},{m}_{1}$ can be adjusted so that $\sigma =\hslash$.

We recall (15), $\bar{V}={V}_{1}+{\bar{V}}_{2}$, where ${V}_{1}=-{{Gm}}_{0}{m}_{1}/r$, $r=| Y|$ and ${\bar{V}}_{2}$ is the time average of ${V}_{2}=-{{Gm}}_{0}{m}_{2}/{r}_{2}$, ${r}_{2}=| Y-\gamma (t)|$. It is shown in this section, ${\bar{V}}_{2}$ is given by (48) which has a form similar to V₁. This enables solving (16).

The solution to (29) is done in two steps. In the first step, we solve (29) in the absence of gravitational perturbations due to P₂ for ${m}_{2}=0$, where ${\bar{V}}_{2}=0$. Thus, in this case $\bar{V}={V}_{1}$. It is then solved for ${m}_{2}\gt 0$ where ${\bar{V}}_{2}$ is non-zero. (Since the form of ${\bar{V}}_{2}$ is shown to have a form analogous to V₁, we can modify the solution obtained for V₁ for the addition of ${\bar{V}}_{2}$.)

We explicitly solve (29), with ${m}_{2}=0$, by separation of variables. This is for the two-body motion of P₀ about P₁. It is first solved for three-dimensions, then restricted to the planar case studied in this paper for the macroscopic scale. The three-dimensional solution is referred to when discussing solutions in the quantum scale. (29) is transformed to spherical coordinates, $r,\phi ,\theta$. It is assumed the solution is of the form,

$$\begin{eqnarray}&&{\rm{\Psi }}=R(r)Y(\phi ,\theta ).\end{eqnarray} \tag{ 30 }$$

$r\geqslant 0$, and ϕ is the angle relative to the Y₁-axis, $0\leqslant \phi \leqslant 2\pi$. θ is the angle relative to the Y₃-axis, $0\leqslant \theta \leqslant \pi$. $| {\rm{\Psi }}{| }^{2}$ is the probability of finding P₀ at distance r from P₁.

The solution of (29) for ${\bar{V}}_{2}=0$ follows the method described in [28] for the case of an electron in the Hydrogen atom moving about the nucleus. This is a standard approach used in solving the classical Schrödinger equation in quantum mechanics found in many references. There are some minor modifications. The Coulomb potential is used in [28] for V₁ and here we are using the gravitational potential between two particles ${P}_{0},{P}_{1};$ however, they are of the same form, both proportional to ${r}^{-1}$, ${r}^{2}={Y}_{1}^{2}+{Y}_{2}^{2}+{Y}_{3}^{2}$. Instead of the proportionality term of ${{Gm}}_{0}{m}_{1}$, the Coulomb potential has the term, ${{Ze}}^{2}{c}_{1}/(4\pi {\epsilon }_{0})$, where Z is the atomic number, Z = 1 for Hydrogen, e is the charge of the electron and the charge of the nucleus, P₀ and an atomic nucleus, P₁, ${\epsilon }_{0}$ is the permittivity of vacuum, $\hslash$ is replaced by σ. The reduced mass $\nu ={m}_{0}{m}_{1}/({m}_{0}+{m}_{1})$ is defined for either the gravitational or Coulomb modeling. When referring to [28], one replaces ${e}^{2}/(4\pi {\epsilon }_{0})$ by ${{Gm}}_{0}{m}_{1}$.

When solving (29) by separation of variables, (30) is substituted into (29), obtaining differential equations for $R(r)$ and $Y(\phi ,\theta )$,

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}R}{{{dr}}^{2}}+2{r}^{-1}\displaystyle \frac{{dR}}{{dr}}+[2\nu {\sigma }^{-2}(E+{{Gm}}_{0}{m}_{1}{r}^{-1})-\alpha {r}^{-2}]R=0,\end{eqnarray} \tag{ 31 }$$

$$\begin{eqnarray}&&{\left(\sin \theta \right)}^{-1}\displaystyle \frac{\partial }{\partial \theta }\left(\sin \theta \displaystyle \frac{\partial Y}{\partial \theta }\right)+{\left(\sin \theta \right)}^{-2}\displaystyle \frac{{\partial }^{2}Y}{\partial {\phi }^{2}}+\alpha Y=0.\end{eqnarray} \tag{ 32 }$$

α is a separation constant.

(32) is solved first, yielding spherical harmonics. Using separation of variables, solutions are obtained in the form, $Y(\phi ,\theta )={\rm{\Phi }}(\phi ){\rm{\Theta }}(\theta )$. This gives the differential equations,

$$\begin{eqnarray}&&{{\rm{\Theta }}}^{-1}\sin \theta \displaystyle \frac{d}{d\theta }\left(\sin \theta \displaystyle \frac{d{\rm{\Theta }}}{d\theta }\right)+\alpha {\sin }^{2}\theta -\beta =0,\end{eqnarray} \tag{ 33 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}{\rm{\Phi }}}{d{\phi }^{2}}+\beta {\rm{\Phi }}=0,\end{eqnarray} \tag{ 34 }$$

where β is a separation constant [28].

(34) gives the solution,

$$\begin{eqnarray}&&{\rm{\Phi }}(\phi )\equiv {{\rm{\Phi }}}_{{m}_{l}}(\phi )={\left(1/2\pi \right)}^{1/2}{{\rm{e}}}^{{{im}}_{l}\phi },\end{eqnarray} \tag{ 35 }$$

$\beta ={m}_{l}^{2},{m}_{l}=0,\pm 1,\pm 2,\ldots$, ${i}^{2}=-1$. ${\rm{\Phi }}(\phi )$ varies in the ${Y}_{1},{Y}_{2}$-plane.

The solution of (33) follows by setting $x=\cos \theta$, $G(x)\equiv {\rm{\Theta }}(\cos x)$ transforming (33) into an associated Legendre type differential equation,

$$\begin{eqnarray}&&(1-{x}^{2})\displaystyle \frac{{d}^{2}G}{{{dx}}^{2}}-2x\displaystyle \frac{{dG}}{{dx}}+\left(\alpha -{m}^{2}{\left(1-{x}^{2}\right)}^{-1}\right)G=0,\end{eqnarray} \tag{ 36 }$$

$\alpha =l(l+1),l=| {m}_{l}| ,| {m}_{l}| +1,| {m}_{l}| +2,\ldots$ [28]. l varies between $\pm | {m}_{l}|$. The solutions of (36) are given by associated Legendre polynomials ${P}_{l,| {m}_{l}| }(x)$. (see [29] for tables of these polynomials) The solutions of (33) are given by ([28], page 527),

$$\begin{eqnarray}&&{{\rm{\Theta }}}_{l,{m}_{l}}(\theta )={\left[\left(\displaystyle \frac{2l+1}{2}\right)\displaystyle \frac{1-| {m}_{l}| !}{1+| {m}_{l}| !}\right]}^{1/2}{P}_{l,| {m}_{l}| }(\cos \theta ).\end{eqnarray} \tag{ 37 }$$

It is remarked that in the two-dimensional problem, with coordinates $({Y}_{1},{Y}_{2})$, $\theta =\pi /2$. In this case, there is no variation with respect to θ and Θ is only defined at $\theta =\pi /2$.

The solution $Y(\phi ,\theta )$ is given by the spherical harmonics ${Y}_{l,{m}_{l}}={{\rm{\Phi }}}_{{m}_{l}}(\phi ){{\rm{\Theta }}}_{l,{m}_{l}}(\theta )$. To solve (31), set $u={Rr}$. (31) becomes,

$$\begin{eqnarray}&&\displaystyle \frac{{d}^{2}u}{{{dr}}^{2}}+(\tilde{a}{r}^{-1}-{{br}}^{-2})u={\lambda }^{2}u,\end{eqnarray} \tag{ 38 }$$

where ${\lambda }^{2}=2\nu | E| {\sigma }^{-2}$, $\tilde{a}=2\nu {\sigma }^{-2}{{Gm}}_{0}{m}_{1}$, $b=l(l+1)$. It is verified that solving this differential equation yields the solution of (31),

$$\begin{eqnarray}&&R\equiv {R}_{\tilde{n},l}(r)=-\left[\displaystyle \frac{2}{\tilde{n}}\left(\displaystyle \frac{(\tilde{n}-l-1)!}{2\tilde{n}{[(\tilde{n}+l)!]}^{3}}\right)\right]{\rho }^{l}{L}_{\tilde{n}+l}^{1l+1}(\rho ){{\rm{e}}}^{-\rho /2},\end{eqnarray} \tag{ 39 }$$

where, $\tilde{n}=1,2\,\ldots$, $l=1,\,\ldots ,\,\tilde{n}-1$, $\rho =(2/(\tilde{n}a))r$, $a={\left(\nu {{Gm}}_{0}{m}_{1}\right)}^{-1}{\sigma }^{2}$, and ${L}_{j}^{i}$ are the associated Laguerre polynomials ([28], page 8, table 3.2), and

$$\begin{eqnarray}&&E\equiv {\hat{E}}_{\tilde{n}}=-\displaystyle \frac{2\nu {\pi }^{2}{\left({{Gm}}_{0}{m}_{1}\right)}^{2}}{{\sigma }^{2}{\tilde{n}}^{2}}.\end{eqnarray} \tag{ 40 }$$

E can be written as,

$$\begin{eqnarray}&&{\hat{E}}_{\tilde{n}}=-\displaystyle \frac{4\sigma }{{\tilde{n}}^{2}}.\end{eqnarray} \tag{ 41 }$$

This follows by the identity,

$$\begin{eqnarray}&&2\nu {\pi }^{2}{\tilde{\rho }}^{2}{\sigma }^{-3}=4,\end{eqnarray} \tag{ 42 }$$

where $\tilde{\rho }={{Gm}}_{0}{m}_{1}$.

The solution for E yields quantized values of the energy, which quantizes the gravitational field between P₀, P₁.

The general solution to (29) is given by multiplying (39) with ${Y}_{l,{m}_{l}}$,

$$\begin{eqnarray}&&{\rm{\Psi }}(r,\phi ,\theta )\equiv {{\rm{\Psi }}}_{\tilde{n},{m}_{l},l}={R}_{\tilde{n},l}(r){{\rm{\Phi }}}_{{m}_{l}}(\phi ){{\rm{\Theta }}}_{l,{m}_{l}}(\theta ).\end{eqnarray} \tag{ 43 }$$

$\tilde{n},l,{m}_{l}$ are the quantum numbers associated with Ψ. $\tilde{n}$ is called the principle quantum number and is independent of $l,{m}_{l}$. It specifies the energy value and limits the value of l. The quantum numbers $l,{m}_{l}$ occur by consideration of the spherical harmonics.

We compute the probability distribution function, F, of locating P₀ relative to P₀ at a given point $(r,\phi ,\theta )$. Since we are currently considering the masses and the relative distances to be macroscopic, this probability is used to measure the location of a macroscopic particle, and not as a wave as is done in the quantum scale, that is considered later in this section. By definition, $F(r,\phi ,\theta )=| {{\rm{\Psi }}}_{\tilde{n},{m}_{l},l}(r,\phi ,\theta ){| }^{2}$, depending on the quantum numbers.

It is more convenient to compute the probability at a given radial distance r, independent of $\phi ,\theta$. It is labeled $P(r)$.

It is verified that $P(r)={R}^{2}{r}^{2}$. This is valid in the two-dimensional case as well for $({Y}_{1},{Y}_{2})$.

As an example, calculate $P(r)$ at the lowest energy value corresponding to $\tilde{n}=1$, $l=0$.

$$\begin{eqnarray}&&P(r)=4{a}^{-3}{r}^{2}{{\rm{e}}}^{-2r/a}.\end{eqnarray} \tag{ 44 }$$

(see [28], table 3.2, where a is given in the Hydrogen atom case). It is verified that $P(r)$ has a maximum at $r=a$, with $P(0)=0$ and where $P(r)\to 0$ as $r\to \infty$. It yields a curve $\{(r,P(r))| r\in [0,\infty ]\}$ analogous to the Hydrogen atom case (see [28], figure 3.20). The numerical values of $P(r)$ will differ from the Hydrogen atom case, since in the gravitational case $a={\left(\nu {{Gm}}_{0}{m}_{1}\right)}^{-1}{\sigma }^{2}$.

The maximum of the distribution function at $r=a$ says that P₀, as a macroscopic body, has the highest probability of being located at this distance. In the case of the Hydrogen atom, where P₀ has wave-particle duality, this distance corresponds to the Bohr radius, which is the most probable location to find an electron in general, referred to as the $1s$-orbital (s denotes l = 0).

In the same way, $P(r)$ can be computed for $\tilde{n}=1,2\ldots ,\hspace{.05in}l=0,1,2,\,\ldots ,\,\tilde{n}-1$, which determines most probable radial locations for P₀ to be located.

If one considers computing $F(r,\phi ,\theta )=| {{\rm{\Psi }}}_{\tilde{n},{m}_{l},l}(r,\phi ,\theta ){| }^{2}$ over $\tilde{n},l,{m}_{l}$, where $-| {m}_{l}| \leqslant l\leqslant | {m}_{l}|$, then one obtains precise regions about P₁ where P₀ is most probable to be located. These are well known in the case of the Hydrogen atom [28]. It is remarkable these are observed to occur. In the gravitational case considered in this paper, the regions will have a similar geometry, but with different scaling.

The solution of (29) has been obtained in three dimensions for ${m}_{2}=0$. We solve this for ${m}_{2}\gt 0$, ${m}_{2}\ll {m}_{1}$, and then reduce to the planar case in order to compare to the planar restricted three-body problem in the macroscopic scale.

When the gravitational perturbation due to P₂ is included, the previous results are obtained with a small perturbation. It is also seen that the frequencies ${\omega }_{1}(\tilde{n})$ correspond to the subset, ${\mathfrak{U}}$, of the resonant family ${\mathfrak{F}}$.

Three-body potential

The previous analysis can be done for a more general three-body potential by taking into account the gravitational perturbation due to P₂. We do this by using an averaged potential, ${\bar{V}}_{2}$, obtained from the potential V₂, due to the gravitational interaction of ${P}_{0},{P}_{2}$. This yields the three-body potential, $\bar{V}={V}_{1}+{\bar{V}}_{2}$ that approximates $\hat{V}={V}_{1}+{V}_{2}$. This is done as follows,

We do this analysis in three-dimensional intertial coordinates, $Y=({Y}_{1},{Y}_{2},{Y}_{3})$, centered at P₁, P₂ moves about P₁ on a circular orbit of radius β, and angular frequency ω, in the $({Y}_{1},{Y}_{2})$-plane, $\gamma (t)=\beta (\cos \omega t,\sin \omega t,0)$, β is a constant. The potential for P₀ due to the perturbation of P₂ is given by

$$\begin{eqnarray}&&\hat{V}={V}_{1}+{V}_{2}=\hspace{.08in}-\displaystyle \frac{{{Gm}}_{0}{m}_{1}}{r}-\displaystyle \frac{{{Gm}}_{0}{m}_{2}}{{r}_{2}},\end{eqnarray} \tag{ 45 }$$

where $r=| Y| ,{r}_{2}=| Y-\gamma (t)|$. We can write r₂ as

$$\begin{eqnarray*}&&{r}_{2}=\sqrt{{r}^{2}+{\beta }^{2}-2\beta ({Y}_{1}\cos \omega t+{Y}_{2}\sin \omega t)},\end{eqnarray*}$$

We consider V₂ and take the average of it over one cycle of P₂, where $t\in [0,2\pi /\omega ]$,

$$\begin{eqnarray}&&{\bar{V}}_{2}=-\displaystyle \frac{{{Gm}}_{0}{m}_{2}\omega }{2\pi }{\int }_{0}^{\tfrac{2\pi }{\omega }}\displaystyle \frac{{dt}}{\sqrt{{r}^{2}+{\beta }^{2}-2\beta ({Y}_{1}\cos \omega t+{Y}_{2}\sin \omega t)}}.\end{eqnarray} \tag{ 46 }$$

This averaged potential term is an approximation to V₂, representing the average value of V₂ felt by P₀ at a point $({Y}_{1},{Y}_{2},{Y}_{3})$ over the circular orbit of P₂ about P₁ in the $({Y}_{1},{Y}_{2})$-plane. It is advantageous to use since it eliminates the time dependence in V₂, and as we'll show, can be written so that it approximately takes the form of V₁. This implies we can solve (29) as before, with minor modifications. Approximating V₂ in this way yields ${\bar{V}}_{2}$.

Expressing ${Y}_{1},{Y}_{2}$ in polar coordinates, ${Y}_{1}=r\cos \theta ,{Y}_{2}=r\sin \theta$, and making a change of the independent variable, t, $\phi =\omega t$, we obtain

$$\begin{eqnarray}&&{\bar{V}}_{2}=-\displaystyle \frac{{{Gm}}_{0}{m}_{2}}{2\pi }{\int }_{0}^{2\pi }\displaystyle \frac{d\phi }{\sqrt{{r}^{2}+{\beta }^{2}-2\beta r\cos (\phi -\theta )}}.\end{eqnarray} \tag{ 47 }$$

${\bar{V}}_{2}$ is simplified by considering three cases, $r\lt \beta$, and $r\gt \beta$, $r=\beta$, and expanding ${\bar{V}}_{2}$ as a binomial series.

We prove,

Summary A. The general three-body potential $\hat{V}={V}_{1}+{V}_{2}$, (12), for P₀ can be approximated by replacing V₂, due to the perturbation of P₂, with the averaged potential ${\bar{V}}_{2}$, (47). ${\bar{V}}_{2}$ can be written as,

$$\begin{eqnarray}{\bar{V}}_{2}=\left\{\begin{array}{ll}-{{Gm}}_{0}{m}_{2}{r}^{-1}+{ \mathcal O }({m}_{0}{m}_{2}), & r\gt \beta \\ -{{Gm}}_{0}{m}_{2}{\beta }^{-1}+{ \mathcal O }({m}_{0}{m}_{2}), & r\lt \beta \\ -{{Gm}}_{0}{m}_{2}\displaystyle \frac{1}{\sqrt{2}}{r}^{-1}+{ \mathcal O }({m}_{0}{m}_{2}), & r=\beta .\end{array}\right.\end{eqnarray} \tag{ 48 }$$

The quantized energy, ${\hat{E}}_{\tilde{n}}$, for the approximated three-body potential $\bar{V}={V}_{1}+{\bar{V}}_{2}$ is given by,

$$\begin{eqnarray}{\hat{E}}_{\tilde{n}}=\left\{\begin{array}{ll}-4\sigma {\tilde{n}}^{-2}(1+\mu +{\mu }^{2})+{ \mathcal O }({m}_{0}{m}_{2}), & r\gt \beta \\ -4\sigma {\tilde{n}}^{-2}-{{Gm}}_{0}{m}_{2}{\beta }^{-1}+{ \mathcal O }({m}_{0}{m}_{2}), & r\lt \beta \\ -4\sigma {\tilde{n}}^{-2}(1+\displaystyle \frac{1}{\sqrt{2}}\mu +\displaystyle \frac{1}{2}{\mu }^{2})+{ \mathcal O }({m}_{0}{m}_{2}), & r=\beta \end{array}\right.\end{eqnarray} \tag{ 49 }$$

which reduces to (41) for ${m}_{2}=0$, and where $\mu ={m}_{2}/{m}_{1}$.

From the form of σ, (49) implies,

$$\begin{eqnarray}&&{\hat{E}}_{\tilde{n}}=-4\sigma {\tilde{n}}^{-2}+{ \mathcal O }({m}_{0}{m}_{2}).\end{eqnarray} \tag{ 50 }$$

Similarly,

Summary B. 

$$\begin{eqnarray}&&{R}_{\tilde{n},l}(r)=-\left[\displaystyle \frac{2}{\tilde{n}}\left(\displaystyle \frac{(\tilde{n}-l-1)!}{2\tilde{n}{[(\tilde{n}+l)!]}^{3}}\right)\right]{\rho }^{l}{L}_{\tilde{n}+l}^{1l+1}(\rho ){{\rm{e}}}^{-\rho /2}+{ \mathcal O }({m}_{0}{m}_{2}),\end{eqnarray} \tag{ 51 }$$

for $r\geqslant \beta ,r\lt \beta$. The probability distribution function is generalized to,

$$\begin{eqnarray}&&P(r)={R}^{2}(r){r}^{2}+{ \mathcal O }({m}_{0}{m}_{2}).\end{eqnarray} \tag{ 52 }$$

When adding the gravitational perturbation due to P₂ represented by ${\bar{V}}_{2}$, one obtains smooth dependence on this term in all the calculations. This proves summary B.

Equivalence of solutions of the modified Schrödinger equation with the family ${\mathfrak{F}}$ of the three-body problem

The two-dimensional case is now considered to compare the solutions of the modified Schrödinger equation, (29), to the family of solutions ${\mathfrak{F}}$ of the planar restricted three-body problem.

We set ${Y}_{3}=0$, $\theta =\pi /2$. The quantized energy ${\hat{E}}_{\tilde{n}}$, (41), for (29) of the two-body motion of P₀ about P₁, with ${m}_{2}=0$, is not defined in the same way as the two-body Kepler energy $\tilde{{E}_{1}}$, (10). ${\hat{E}}_{\tilde{n}}$ is computed from the modified Schrödinger equation and $\tilde{{E}_{1}}$ is computed for the Kepler problem for general elliptic motion of P₀ about P₁. These are different expressions. However, they both represent the energy of P₀ for the two-body gravitational potential. Also, ${\hat{E}}_{\tilde{n}}$ is quantized and $\tilde{{E}_{1}}$ is not quantized.

A key observation of this paper is that when one solves for the Kepler frequency ${\omega }_{1}$ in (10) as a function of $\tilde{{E}_{1}}$ and substitutes ${\hat{E}}_{\tilde{n}}$ in place of $\tilde{{E}_{1}}$, a simple equation is obtained for ${\omega }_{1}$,

$$\begin{eqnarray}&&{\omega }_{1}{| }_{\tilde{{E}_{1}}={\hat{E}}_{\tilde{n}}}\equiv {\omega }_{1}(\tilde{n})=8{\tilde{n}}^{-3}.\end{eqnarray} \tag{ 53 }$$

It is remarked that this equation is also valid in three-dimensions since (10) is also valid for the three-dimensional Kepler two-body problem.

This follows by noting that (10) implies,

$$\begin{eqnarray}&&{\omega }_{1}={[-{\sigma }^{-1}{\tilde{E}}_{1}]}^{3/2}.\end{eqnarray} \tag{ 54 }$$

This yields (53)

Inclusion of gravitational perturbation of P₂ (${m}_{2}\gt 0$), implies more generally,

$$\begin{eqnarray}&&{\omega }_{1}{| }_{\tilde{{E}_{1}}={\hat{E}}_{\tilde{n}}}\equiv {\omega }_{1}(\tilde{n})=8{\tilde{n}}^{-3}+{ \mathcal O }({m}_{0}{m}_{2}).\end{eqnarray} \tag{ 55 }$$

(53) does not depend on the masses or any other physical parameter. This says substituting the quantized energy from the modified Schrödinger equation, with ${m}_{2}=0$, into the Kepler energy for the frequency, yields a frequency of motion for P₀ moving about P₁ in elliptical orbits that is same for all masses only depending on the wave number $\tilde{n}$. This discretizes the Kepler frequencies.

The discretization of the Kepler frequencies restricts the elliptical two-body motion of P₀. This result becomes relevant in the three-body problem for ${\mathfrak{F}}$ when the gravitational perturbation P₂ is included since it selects a the set of resonance orbits, ${\mathfrak{U}}\subset {\mathfrak{F}}$ (see result C, section 2).

We prove result C from section 2, which we state as ${\omega }_{1}(\tilde{n})$, given by (55), which are the frequencies in the restricted three-body problem corresponding to the modified Schrödinger equation energies, form a subset ${\mathfrak{U}}\subset {\mathfrak{F}}$ of resonance orbits where $m=8,n=\tilde{n}$.

Proof of result C. This is proven by first noting that the circular restricted three-body problem can rescaled so that ${m}_{1}=1-\mu ,{m}_{2}=\mu ,G=1,\beta =1$ where $\mu ={m}_{2}/({m}_{1}+{m}_{2})$ [8]. This scaling does not reduce the generality of the mass values nor β. This scaling implies $\omega =1$. Thus, (6) becomes,

$$\begin{eqnarray}&&{\omega }_{1}(m/n)=(m/n)+{ \mathcal O }(\delta ).\end{eqnarray} \tag{ 56 }$$

A key observation is that this scaling does not effect the leading term $8/{\tilde{n}}^{3}$ of ${\omega }_{1}(\tilde{n})$ given by (55). Thus, after the scaling, subtracting (55) from (56) yields,

$$\begin{eqnarray}&&{\omega }_{1}(m/n)-{\omega }_{1}(\tilde{n})=\displaystyle \frac{m}{n}-\displaystyle \frac{8}{{\tilde{n}}^{3}}+{ \mathcal O }({m}_{0}{m}_{2}).\end{eqnarray} \tag{ 57 }$$

Thus, taking $m=8,n={\tilde{n}}^{3}$ and assuming m₂ is sufficiently small, implies,

$$\begin{eqnarray}&&{\omega }_{1}(m/n)\approx {\omega }_{1}(\tilde{n}),\end{eqnarray} \tag{ 58 }$$

This condition is preserved by rescaling to general ${m}_{1},{m}_{2},\beta$, yielding ${\omega }_{1}(m/n)\approx (8/{\tilde{n}}^{3})\omega$ for ${\mathfrak{U}}$.

From the macro to quantum scale

When ${m}_{0},{m}_{2},{m}_{3}$ are in the macroscopic scale then as described in section 2, in results B and C, the family ${\mathfrak{U}}\subset {\mathfrak{F}}$ of near resonance orbits can be described by the solution (17) of (29).

For the masses in the quantum scale, the family ${\mathfrak{U}}$ are no longer valid. Ψ given by (17) is still valid but now as pure wave solutions. This is summarized in result D, section 2. Thus, Ψ is defined for both macroscopic and quantum scales. In the macroscopic scale, Ψ is interpreted as a probability, whereas in the quantum scale, Ψ is a pure wave solution. The quantized energies ${\hat{E}}_{\tilde{n}}$ are still well defined. ${ \mathcal P }$ is still defined and yields the single domain ${ \mathcal D }$, for $\tilde{n}=1,2,3\ldots$. This is discussed in section 2.

This section is concluded with an analysis of σ, referred to in section 2.

σ = ℏ is satisfied for a one-dimensional algebraic curve (59) in ( m 0 , m 1 ) -space.

Proposition 4.1 This is proven by noting $\sigma =(1/2){\left(2\pi G\right)}^{2/3}{\left.{m}_{0}{m}_{1}({m}_{0}+{m}_{1}\right)}^{-1/3}\lt (1/2){\left(2\pi G\right)}^{2/3}{m}_{0}{m}_{1}{m}_{0}^{-1/3}$. Hence, $\sigma \lt (1/2){\left(2\pi G\right)}^{2/3}{m}_{0}^{2/3}{m}_{1}$. Thus, $\sigma \to 0$ as ${m}_{0},{m}_{1}\to 0$. This implies there exists values of ${m}_{0},{m}_{1}$ such that $\sigma =\hslash$. This is equivalent to the equation,

$$\begin{eqnarray}&&{m}_{0}^{3}{m}_{1}^{3}-\hslash {a}^{-3}({m}_{0}+{m}_{1})=0,\end{eqnarray} \tag{ 59 }$$

$a=(1/2){\left(2\pi G\right)}^{2/3}$. (59) yields a one-dimensional algebraic curve, Γ, in the coordinates ${m}_{0},{m}_{1}$. This proves proposition 4.1.

The relevancy of possible wave motions for $({m}_{0},{m}_{1})$ not on Γ, with ${m}_{0},{m}_{1},{m}_{2}$ in the quantum scale, is not considered in this paper. This is discussed in section 2. Different models are also discussed in section 2.

I would like acknowledge the support of Alexander von Humboldt Stiftung of the Federal Republic of Germany that made this research possible, and the support of the University of Augsburg for his visit from 2018–19. I would like to thank Urs Frauenfelder of the University of Augsburg for many interesting discussions. Research by E.B. was partially supported by NSF grant DMS-1814543. Special thanks to Marian Gidea of Yeshiva University for helpful discussions and figure 2. Also, thanks to David Spergel of Princeton University.

Summary A is proven as follows: the integrand, I, of ${\bar{V}}_{2}$ is given by

$$\begin{eqnarray}&&I=\displaystyle \frac{1}{\sqrt{{r}^{2}+{\beta }^{2}-2\beta r\cos (\phi -\theta )}}=\displaystyle \frac{1}{\sqrt{{r}^{2}+{\beta }^{2}}}\displaystyle \frac{1}{\sqrt{1-\tfrac{2\beta r}{{r}^{2}+{\beta }^{2}}\cos (\phi -\theta )}}.\end{eqnarray} \tag{ 60 }$$

Case 1: $r\gt \beta$

This implies,

$$\begin{eqnarray}&&\displaystyle \frac{1}{\sqrt{{r}^{2}+{\beta }^{2}}}=\displaystyle \frac{1}{r\sqrt{1+\tfrac{{\beta }^{2}}{{r}^{2}}}}=\displaystyle \frac{1}{r}(1+{ \mathcal O }(x)),\end{eqnarray} \tag{ 61 }$$

where $x=\beta /r\lt 1$. This results from expanding the fraction containing the square root into a binomial series. Likewise, we can also expand the first term on the right in (60) in a binomial expansion since $| \cos (\phi -\theta )| \leqslant 1$ and

$$\begin{eqnarray}&&\displaystyle \frac{2\beta r}{{r}^{2}+{\beta }^{2}}\lt 1,\end{eqnarray} \tag{ 62 }$$

yielding

$$\begin{eqnarray}&&\displaystyle \frac{1}{\sqrt{1-\tfrac{2\beta r}{{r}^{2}+{\beta }^{2}}\cos (\phi -\theta )}}=1+{ \mathcal O }(y),\end{eqnarray} \tag{ 63 }$$

where

$$\begin{eqnarray}&&y=\displaystyle \frac{2\beta r}{{r}^{2}+{\beta }^{2}}(\cos (\phi -\theta )),\end{eqnarray} \tag{ 64 }$$

$| y| \lt 1$. Thus, (60) becomes,

$$\begin{eqnarray}&&I=\displaystyle \frac{1}{r}(1+{ \mathcal O }(w)),\end{eqnarray} \tag{ 65 }$$

$| w| \lt 1$, $w=\max \{x,y\}$. Thus, in this case,

$$\begin{eqnarray}&&{\bar{V}}_{2}=-\displaystyle \frac{{{Gm}}_{0}{m}_{2}}{r}(1+{ \mathcal O }(w))=-\displaystyle \frac{{{Gm}}_{0}{m}_{2}}{r}+{ \mathcal O }({m}_{0}{m}_{2}).\end{eqnarray} \tag{ 66 }$$

This implies in the derivation of ${\hat{E}}_{\tilde{n}}$, we proceed as before and replace the numerator ${{Gm}}_{0}{m}_{1}$ of V₁ in (31) with ${{Gm}}_{0}({m}_{1}+{m}_{2})$ and adding ${ \mathcal O }({m}_{0}{m}_{2})$ to this term. This yields,

$$\begin{eqnarray}&&{\hat{E}}_{\tilde{n}}=-\displaystyle \frac{2\nu {\pi }^{2}{\left({{Gm}}_{0}({m}_{1}+{m}_{2}\right)}^{2}}{{\sigma }^{2}{\tilde{n}}^{2}}+{ \mathcal O }({m}_{0}{m}_{2})\end{eqnarray} \tag{ 67 }$$

This can be written as,

$$\begin{eqnarray}&&{\hat{E}}_{\tilde{n}}=-\displaystyle \frac{4\sigma }{{\tilde{n}}^{2}}(1+\mu +{\mu }^{2})+{ \mathcal O }({m}_{0}{m}_{2}),\end{eqnarray} \tag{ 68 }$$

where $\mu ={m}_{2}/{m}_{1}$, and where we have used (42).

Case 2: $r\lt \beta$

This case is done in a similar way as in Case 1. Instead of factoring out ${r}^{-1}$ from (61), we factor out ${\beta }^{-1}$, which yields,

$$\begin{eqnarray}&&\displaystyle \frac{1}{\sqrt{{r}^{2}+{\beta }^{2}}}=\displaystyle \frac{1}{\beta \sqrt{1+\tfrac{{r}^{2}}{{\beta }^{2}}}}=\displaystyle \frac{1}{\beta }(1+{ \mathcal O }(\tilde{x})),\end{eqnarray} \tag{ 69 }$$

where $\tilde{x}=r/\beta \lt 1$. Proceeding as in Case 1, we obtain,

$$\begin{eqnarray}&&I=\displaystyle \frac{1}{\beta }(1+{ \mathcal O }(\tilde{w})),\end{eqnarray} \tag{ 70 }$$

$| \tilde{w}| \lt 1$, $\tilde{w}=\max \{\tilde{x},y\}$. This implies,

$$\begin{eqnarray}&&{\bar{V}}_{2}=-\displaystyle \frac{{{Gm}}_{0}{m}_{2}}{\beta }(1+{ \mathcal O }(\tilde{w}))=-\displaystyle \frac{{{Gm}}_{0}{m}_{2}}{\beta }+{ \mathcal O }({m}_{0}{m}_{2}).\end{eqnarray} \tag{ 71 }$$

Since β is a constant, then in the derivation of ${\hat{E}}_{\tilde{n}}$, we replace E with $E+({{Gm}}_{0}{m}_{2}/\beta )+{ \mathcal O }({m}_{0}{m}_{2})$ in (31) keeping V₁ as was used in the case of ${m}_{2}=0$ in (31). This yields,

$$\begin{eqnarray}&&{\hat{E}}_{\tilde{n}}=-\displaystyle \frac{2\nu {\pi }^{2}{\left({{Gm}}_{0}{m}_{1}\right)}^{2}}{{\sigma }^{2}{\tilde{n}}^{2}}-\displaystyle \frac{{{Gm}}_{0}{m}_{2}}{\beta }+{ \mathcal O }({m}_{0}{m}_{2})\end{eqnarray} \tag{ 72 }$$

This can be reduced to,

$$\begin{eqnarray}&&{\hat{E}}_{\tilde{n}}=-\displaystyle \frac{4\sigma }{{\tilde{n}}^{2}}-\displaystyle \frac{{{Gm}}_{0}{m}_{2}}{\beta }+{ \mathcal O }({m}_{0}{m}_{2}).\end{eqnarray} \tag{ 73 }$$

Case 3: $r=\beta$

In this final case,

$$\begin{eqnarray}&&\displaystyle \frac{1}{\sqrt{{r}^{2}+{\beta }^{2}}}=\displaystyle \frac{1}{r\sqrt{1+\tfrac{{r}^{2}}{{\beta }^{2}}}}=\displaystyle \frac{1}{\sqrt{2}r}.\end{eqnarray} \tag{ 74 }$$

Thus,

$$\begin{eqnarray}&&I=\displaystyle \frac{1}{\sqrt{{r}^{2}+{\beta }^{2}}}\displaystyle \frac{1}{\sqrt{1-\tfrac{2\beta r}{{r}^{2}+{\beta }^{2}}\cos (\phi -\theta )}}=\displaystyle \frac{1}{\sqrt{2}r}\displaystyle \frac{1}{\sqrt{1-h(\psi )}},\end{eqnarray} \tag{ 75 }$$

where $h=\cos (\psi )$, $\psi =\phi -\theta$. We assume that P₀ does not collide with P₂, implying $\psi \ne 0,\pm 2j\pi$, $j=1,2,3,\ldots$. Thus, $| h| \lt 1$. We can write I as,

$$\begin{eqnarray}&&I=\displaystyle \frac{1}{\sqrt{2}r}(1+{ \mathcal O }(h)).\end{eqnarray} \tag{ 76 }$$

Hence,

$$\begin{eqnarray}&&{\bar{V}}_{2}=-\displaystyle \frac{{{Gm}}_{0}{m}_{2}}{\sqrt{2}r}+{ \mathcal O }({m}_{0}{m}_{2}).\end{eqnarray} \tag{ 77 }$$

Proceeding as in Case 1,

$$\begin{eqnarray}&&{\hat{E}}_{\tilde{n}}=-\displaystyle \frac{2\nu {\pi }^{2}{\left({{Gm}}_{0}\left({m}_{1}+\tfrac{1}{\sqrt{2}}{m}_{2}\right)\right)}^{2}}{{\sigma }^{2}{\tilde{n}}^{2}}+{ \mathcal O }({m}_{0}{m}_{2})\end{eqnarray} \tag{ 78 }$$

This can be written as,

$$\begin{eqnarray}&&{\hat{E}}_{\tilde{n}}=-\displaystyle \frac{4\sigma }{{\tilde{n}}^{2}}\left(1+\displaystyle \frac{1}{\sqrt{2}}\mu +\displaystyle \frac{1}{2}{\mu }^{2}\right)+{ \mathcal O }({m}_{0}{m}_{2}).\end{eqnarray} \tag{ 79 }$$